Christian Zielinski, PhD

Theoretical and Computational Aspects

of New Lattice Fermion Formulations

Christian Zielinski

Division of Mathematical Sciences

School of Physical and Mathematical Sciences

Nanyang Technological University

2016

arXiv:1703.06364v1 [hep-lat] 18 Mar 2017

T  C A

 N L F F

C Z

Division of Mathematical Sciences

School of Physical and Mathematical Sciences

Nanyang Technological University

A thesis submitted to the Nanyang Technological University

in partial fulﬁllment of the requirement for the degree of

Doctor of Philosophy in the Mathematical Sciences

2016

A

In this work we investigate theoretical and computational aspects of novel lat-

tice fermion formulations for the simulation of lattice gauge theories. The lattice

approach to quantum gauge theories is an important tool for studying quantum

chromodynamics, where it is the only known framework for calculating physi-

cal observables from ﬁrst principles. In our investigations we focus on staggered

Wilson fermions and the related staggered domain wall and staggered overlap for-

mulations. Originally proposed by Adams, these new fermion discretizations bear

the potential to reduce the computational costs of state-of-the-art Monte Carlo

simulations. Staggered Wilson fermions combine aspects of both staggered and

Wilson fermions while having a reduced number of fermion doublers compared to

usual staggered fermions. Moreover, they can be used as a kernel operator for the

domain wall fermion construction with potentially signiﬁcantly improved chiral

properties and for the overlap operator with its exact chiral symmetry. This allows

the implementation of chirality on the lattice in a controlled manner at potentially

signiﬁcantly reduced costs. The practical potential and limitations of these new

lattice fermions are also critically discussed.

A

Over the course of the four years of my Ph.D. I had the pleasure to work with

many brilliant people in my ﬁeld. My special thanks go to Dr. David H. Adams

for guiding me in my ﬁrst three years, for plenty of insightful discussions, for all

the constructive feedback and his conﬁdence in my abilities. I also want to thank

our collaborator Asst. Prof. Daniel Nogradi from the Eötvös Loránd University for

patiently answering many questions by email. My thanks also go to Assoc. Prof.

Wang Li-Lian for his supervision in my ﬁnal year. Both in private discussions and

in his excellent lectures I learned a lot about the numerical foundations for my

research.

At the Nanyang Technological University I want to thank my lecturers, who

all contributed in widening my horizons. Moreover, I had the pleasure working

together with many great colleagues and having plenty of interesting discussions

with friends. My special thanks go here to Dr. Reetabrata Har, Jia Yiyang, Sai

Swaroop Sunku, Andrii Petrashyk, Christian Engel, Ido David Polak, Nguyen Thi

Phuc Tan, Liu Zheng, Chen Penghua and Lim Kim Song. During my time at the

Nanyang Technological University I got to know many great people, for whom I

am very thankful.

I spent the ﬁnal year of my Ph.D. in the Theoretical Particle Physics Group at

the University of Wuppertal (Bergische Universität Wuppertal). Here my special

thanks go to PD Dr. Christian Hoelbling and Prof. Zoltán Fodor, who made my stay

in Wuppertal possible. I greatly enjoyed working together with Christian Hoelbling,

as not only did I learn a lot in our discussions, but he was also always encouraging

and highly motivated. My thanks also go to PD Dr. Stephan Dürr, who made a lot

of great suggestions and gave helpful feedback on our projects and manuscripts,

and to Prof. Ting-Wai Chiu at the National Taiwan University for answering many

questions by email. I was lucky to share my oﬃce with two great colleagues, Dr.

Attila Pásztor and Dr. Thomas Rae. My stay in Wuppertal was very enjoyable and

productive and my thanks go also to the rest of the group.

I want to thank PD Dr. Christian Hoelbling, Assoc. Prof. Wang Li-Lian, Jia Yiyang,

Christian Engel and Pooi Khay Chong for providing feedback on various parts of

a draft of this thesis. I also thank the three anonymous examiners of this thesis

for their constructive criticisms and valuable comments. Furthermore, I want to

acknowledge ﬁnancial support from the Singapore International Graduate Award

(SINGA) and Nanyang Technological University.

Finally, I want to express my utmost gratitude to my family and my beloved wife

for their unconditional support over all these years.

vii

L  

As part of this Ph.D. the following work was published:

C. Hoelbling and C. Zielinski, “Staggered domain wall fermions,” PoS

LAT-

TICE2016 (2016) 254, arXiv:1609.05114 [hep-lat]

2. C. Hoelbling and C. Zielinski, “Spectral properties and chiral symmetry vio-

lations of (staggered) domain wall fermions in the Schwinger model,” Phys.

Rev. D94 no. 1, (2016) 014501, arXiv:1602.08432 [hep-lat]

D. H. Adams, R. Har, Y. Jia, and C. Zielinski, “Continuum limit of the axial

anomaly and index for the staggered overlap Dirac operator: An overview,”

PoS LATTICE2013 (2014) 462, arXiv:1312.7230 [hep-lat]

D. H. Adams, D. Nogradi, A. Petrashyk, and C. Zielinski, “Computational

eﬃciency of staggered Wilson fermions: A ﬁrst look,” PoS

LATTICE2013

(2014) 353, arXiv:1312.3265 [hep-lat]

Note: In lattice ﬁeld theory authors are traditionally ordered by name.

L  

Parts of this work have been presented by the author at the following venues:

2016 Workshop of the Collaborative Research Center Transregio SFB/TR-55,

University of Wuppertal, Wuppertal, Germany

2016 Spring Meeting of the German Physical Society,

University of Hamburg, Hamburg, Germany

2015 Invited Talk at the Theoretical Particle Physics Seminar,

University of Wuppertal, Wuppertal, Germany

2015 33

International Symposium on Lattice Field Theory,

Kobe International Conference Center, Kobe, Japan

2015 Conference on 60 Years of Yang-Mills Gauge Field Theories,

Institute of Advanced Studies, Nanyang Technological University, Singapore

2015 9

International Conference on Computational Physics,

National University of Singapore, Singapore

2014 5

Singapore Mathematics Symposium,

National University of Singapore, Singapore

2014 32

International Symposium on Lattice Field Theory,

Columbia University, New York City, United States

CONTENTS

Abstract v

Acknowledgements vii

List of publications ix

List of presentations xi

1 Introduction 1

1.1 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Original contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Gauge theories in a nutshell 5

2.1 Gauge theories in the continuum . . . . . . . . . . . . . . . . . . . 5

2.1.1 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.3 Chiral symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.4 Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Lattice gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Wick rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Naïve lattice fermions . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Fermion doubling . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.4 The Nielsen-Ninomiya No-Go theorem . . . . . . . . . . . . . 15

2.2.5 Coupling to gauge ﬁelds . . . . . . . . . . . . . . . . . . . . . 15

2.2.6 Wilson fermions . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.7 Staggered fermions . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.8 Staggered Wilson fermions . . . . . . . . . . . . . . . . . . . . 21

2.2.9 Gauge ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . 22

3 Staggered Wilson fermions 25

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Symmetries of staggered fermions . . . . . . . . . . . . . . . . . . . 26

3.3 Flavored mass terms . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Adams’ mass term . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

xiii

Contents

3.4.4 Aoki phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5 Hoelbling’s mass term . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.6 Generalized mass terms . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.6.1 The four-dimensional case . . . . . . . . . . . . . . . . . . . . 35

3.6.2 The d-dimensional case . . . . . . . . . . . . . . . . . . . . . 37

4 Computational efficiency 39

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Theoretical estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.1 Number of iterations . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.2 Cost of matrix-vector multiplications . . . . . . . . . . . . . . 42

4.3 Numerical study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3.2 Overall speedup . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 Arithmetic intensities . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4.1 Wilson fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4.2 Staggered fermions . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4.3 Staggered Wilson fermions . . . . . . . . . . . . . . . . . . . . 51

4.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 Pseudoscalar meson spectrum 55

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2 Pseudoscalar mesons . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6 Staggered overlap fermions 63

6.1 Neuberger’s overlap fermions . . . . . . . . . . . . . . . . . . . . . . 64

6.1.1 The Ginsparg-Wilson relation . . . . . . . . . . . . . . . . . . 64

6.1.2 The overlap operator . . . . . . . . . . . . . . . . . . . . . . . 66

6.2 Staggered overlap fermions . . . . . . . . . . . . . . . . . . . . . . . 67

6.3 The index theorem and the axial anomaly . . . . . . . . . . . . . . . 68

6.4 Continuum limit of the index and axial anomaly . . . . . . . . . . . 69

6.4.1 The index and axial anomaly . . . . . . . . . . . . . . . . . . . 70

6.4.2 Continuum limit of the index density . . . . . . . . . . . . . . 71

7 Spectral properties 75

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.2 Free-ﬁeld case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.2.1 Wilson fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.2.2 Staggered Wilson fermions . . . . . . . . . . . . . . . . . . . . 78

7.3 Quenched quantum chromodynamics . . . . . . . . . . . . . . . . . 79

xiv

Contents

7.3.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7.3.2 Eigenvalue spectra . . . . . . . . . . . . . . . . . . . . . . . . 81

7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

8 Staggered domain wall fermions 85

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8.2 Kernel operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

8.3 Domain wall fermions . . . . . . . . . . . . . . . . . . . . . . . . . 87

8.3.1 Standard construction . . . . . . . . . . . . . . . . . . . . . . 88

8.3.2 Boriçi’s construction . . . . . . . . . . . . . . . . . . . . . . . 89

8.3.3 Optimal construction . . . . . . . . . . . . . . . . . . . . . . 90

8.3.4 Staggered formulations . . . . . . . . . . . . . . . . . . . . . . 92

8.4 Eﬀective Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . 93

8.4.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

8.4.2 The N

→ ∞ limit . . . . . . . . . . . . . . . . . . . . . . . . . 95

8.4.3 Approximate sign functions . . . . . . . . . . . . . . . . . . . 96

8.5 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

8.5.1 Setting the domain wall height . . . . . . . . . . . . . . . . . . 98

8.5.2 Eﬀective mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

8.5.3 Normality and Ginsparg-Wilson relation . . . . . . . . . . . . 99

8.5.4 Topological charge . . . . . . . . . . . . . . . . . . . . . . . . 100

8.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.6.1 Free-ﬁeld case . . . . . . . . . . . . . . . . . . . . . . . . . . 101

8.6.2 U

(

)

gauge ﬁeld case . . . . . . . . . . . . . . . . . . . . . . . 105

8.6.3 Smearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

8.6.4 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.6.5 Spectral ﬂow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8.6.6 Approaching the continuum . . . . . . . . . . . . . . . . . . . 112

8.7 Quenched quantum chromodynamics . . . . . . . . . . . . . . . . . 119

8.8 Optimal weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

9 Conclusions 123

9.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

9.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Bibliography 125



C 

I

Gauge theories are some of the most fundamental building blocks in modern

physics. In a gauge theory, physical observables are invariant under transforma-

tions of some fundamental ﬁelds. This property is referred to as gauge symmetry.

Currently all known fundamental forces in nature can be described as classical

or quantum gauge theories. General relativity is invariant under arbitrary coordi-

nate transformations, quantum electrodynamics (QED) is an Abelian gauge theory,

while quantum chromodynamics (QCD) and the electroweak interaction are of the

Yang-Mills type.

The generally accepted theory of the strong nuclear force is quantum chromody-

namics, which describes the interactions between quarks and gluons. The strong

nuclear force binds quarks and gluons to form hadrons, among them nuclear par-

ticles such as the proton and neutron. In the case of quantum chromodynamics,

we deal with a non-Abelian gauge theory with gauge group

(

)

. The associated

force carrier is the so-called gluon, the associated charge is called color. Quan-

tum chromodynamics is the primary setting of this thesis and has some peculiar

properties, in particular conﬁnement and asymptotic freedom.

Conﬁnement refers to the property that we can only observe color-singlet states

and, thus, quarks and gluons cannot propagate freely. When quarks are pulled

apart, the energy of the gluonic ﬁeld in between them increases until a new quark

pair is created. While conﬁnement is phenomenologically well-established, no

rigor mathematical proof of conﬁnement is known. In fact, conﬁnement would

follow from proving the existence of a positive mass gap in quantum Yang-Mills

theory, which is one of the famous Millennium Problems of the Clay Mathematics

Institute.

Asymptotic freedom, on the other hand, refers to the fact that strong nuclear

interactions become weak at high energy scales. In experiments, such as heavy ion

collisions, one can then ﬁnd a new state of matter, the quark-gluon plasma. Due

to the extremely high energy densities in the quark-gluon plasma, quarks become

deconﬁned.

1 Introduction

In general, the non-Abelian nature of quantum chromodynamics makes its

treatment with analytical methods diﬃcult. While in certain regimes quantum

chromodynamics can be treated by perturbative methods much in the same way

quantum electrodynamics is treated, in the low-energy regime of bound hadrons

the coupling constant of the strong nuclear interaction is of order

(

)

and pertur-

bation theory is rendered inapplicable.

An alternative and non-perturbative approach is the framework of lattice quan-

tum chromodynamics. Originally proposed by Wilson in Refs. [

], one discretizes

the theory by replacing continuous space-time by a hypercubic space-time grid,

the so-called lattice. The resulting numerical formulation can, then, be simulated

on computers. As the resulting lattices usually have a very large number of sites, in

particular in the case of four or more space-time dimensions, extracting accurate

predictions from the theory is computationally extremely challenging. Enormous

amounts of computational resources are, thus, required for lattice ﬁeld theoretical

simulations and the interest in ﬁnding more eﬃcient and accurate lattice formu-

lations is high. As a consequence the subject has been and remains a very active

topic of research. The vast numbers of proposed lattice formulations suggest that

no “optimal formulation” is known, but that some are more suitable than others

depending on the application.

As part of these ongoing research eﬀorts, in this thesis we study theoretical and

computational aspects of staggered Wilson, staggered overlap and staggered do-

main wall fermions as proposed by Adams. These novel lattice fermion discretiza-

tions have the potential of reducing computational costs in numerical simulations

and allow the generalization of the overlap and domain wall fermion formulation

to the case of a staggered kernel. While the construction of these lattice fermi-

ons are very interesting from a theoretical point of view, we also evaluate them

under practical aspects and try to understand in what setting their use can be

advantageous.

Although the lattice approach is traditionally concerned with quantum chromo-

dynamics, it is also used to study a large spectrum of other quantum ﬁeld theories.

In this thesis, for example, we consider both four-dimensional quantum chromody-

namics and the Schwinger model, i.e. two-dimensional quantum electrodynamics.

1.1 Outline of this thesis

This thesis is structured as follows. In Chapter 2, we give a short review of Dirac

fermions and gauge theories both in the continuum and on the lattice. In Chapter

3, we recapitulate the construction and properties of ﬂavored mass terms and

staggered Wilson fermions. In Chapter 4, we discuss the computational eﬃciency

of staggered Wilson fermions and compare them to the one of Wilson fermions. In

Chapter 5, we discuss the feasibility of spectroscopy calculations with staggered

Wilson fermions and explicitly compute the pseudoscalar meson spectrum. In

Chapter 6, we review the formulation of usual and staggered overlap fermions on

the lattice and analytically evaluate the continuum limit of the index and the axial

anomaly. In Chapter 7, we investigate the eigenvalue spectra of both the Wilson and

staggered Wilson kernel and relate them to the computational eﬃciency of their

corresponding overlap operators. In Chapter 8, we discuss the formulation of usual

and staggered domain wall fermions in detail and compare spectral properties and

chiral symmetry violations in the case of the Schwinger model. Finally, in Chapter

9 we summarize the central results of this thesis and provide an outlook.

1.2 Original contributions

Apart from an extensive review of the lattice fermion formulations considered in

this thesis, there are various novel contributions as listed in the following:



The generalized ﬂavored mass terms for arbitrary mass splittings in arbitrary

even dimensions in Sec. 3.6 as proposed in Ref. [

] (done in collaboration

with C. Hoelbling).



The quantitative analysis of the computational eﬃciency and memory band-

width requirements of the staggered Wilson kernel in Chapter 4 as discussed

in Refs. [

] (done in collaboration with D. H. Adams, D. Nogradi and A.

Petrashyk).



The adaption of staggered spectroscopy methods to the two-ﬂavor case and

the computation of the pseudoscalar meson spectrum with staggered Wilson

fermions in Chapter 5 as presented in Refs. [

] (done in collaboration

with D. H. Adams).

1 Introduction



The analysis of the continuum limit of the index and axial anomaly for the

staggered overlap Dirac operator

in Sec. 6.4 as given in Ref. [

] (done in

collaboration with D. H. Adams, R. Har and Y. Jia).



The discussion of the usual Wilson and staggered Wilson eigenvalue spec-

trum and its connection to the computational eﬃciency of the overlap con-

struction in Chapter 7 as presented in Refs. [

] (done in collaboration with

D. H. Adams).



The study of spectral properties and chiral symmetry violations of various

known and new variants of usual and staggered domain wall fermions in

Chapter 8 as discussed in Refs. [

] (done in collaboration with C. Hoel-

bling).

Some results presented in this thesis have already been published in the literature

as listed on page ix, while others were presented at international conferences as

listed on page xi. Throughout the thesis the author uses the inclusive “we”, to

include both the reader and collaborators in the discussion.

Note here also R. Har’s thesis in Ref. [11], elaborating on this derivation in more detail.



C 

G    

Quantum gauge theories play a central role in modern physics. Their deﬁning prop-

erty is that the action is invariant under a continuous group of local transformations.

The standard model can be understood as a gauge theory with non-Abelian gauge

group

(

)

× SU

(

)

(

)

. It includes quantum chromodynamics (QCD) with

gauge group

(

)

and the electroweak sector with gauge group

(

)

(

)

. The

electroweak interaction describes a uniﬁed theory of the weak and electromagnetic

interactions.

Throughout this thesis we are dealing primarily with two speciﬁc gauge theories.

The ﬁrst being quantum chromodynamics, which is a Yang-Mills theory with non-

Abelian symmetry group

(

)

. The other one is quantum electrodynamics, which

is an Abelian gauge theory with gauge group U

(

)

In this chapter, we review the formulation of gauge theories in the continuum in

Sec. 2.1 and on the lattice in Sec. 2.2. We also brieﬂy discuss the use of Monte Carlo

methods for the simulation of lattice gauge theories in Sec. 2.3. For notational ease,

we only discuss the case of four space-time dimensions. For a detailed discussion

of the continuum case, we refer the reader to the many textbooks in the ﬁeld such

as Refs. [13–15]. For the lattice formulation, see e.g. Refs. [16–19].

2.1 Gauge theories in the continuum

We begin by reviewing some aspects of gauge theories in continuous Minkowski

space-time. Throughout this thesis we are using natural units, where

~ = c = 1 (2.1.1)

with the reduced Planck constant

~ ≡ h/

(

2π

)

and the speed of light

. In this unit

system, energy is the only remaining independent dimension. For converting back

to conventional units, we note that ~c = 197.3 MeV· fm.

2 Gauge theories in a nutshell

In our convention the Minkowski metric tensor reads

µν

= η

µν







−1







(2.1.2)

and we follow Einstein’s sum convention by summing over repeated indices. Greek

indices run over 0, 1, 2, 3, while roman indices are restricted to the spacial compo-

nents 1, 2, 3. Indices of four-vectors can be lowered and raised using the metric

tensor. If x denotes the spatial part of a four-vector, we have



, x



, x

= η

µν



, −x



. (2.1.3)

Furthermore, we introduce the derivative operator as

∂

∂x

. (2.1.4)

In the following, we discuss the basic framework for the quantum ﬁeld theories of

interest in continuous Minkowski space-time. We begin by discussing fermions in

Subsec. 2.1.1 and gauge theories in Subsec. 2.1.2, followed by chiral symmetry in

Subsec. 2.1.3 and the path integral formalism in Subsec. 2.1.4. In our notation and

discussion we follow Ref. [15].

2.1.1 Fermions

In the discussion of fermions we restrict ourselves to the case of spin-

Dirac fer-

mions, such as quarks and electrons. They can be described by the Dirac equation,

a relativistic ﬁrst-order partial diﬀerential equation ﬁrst introduced in Ref. [

]. In

the free-ﬁeld case it is of the form



iγ

∂

− m



(

)

= 0, (2.1.5)

where

denotes the fermion mass and

is a four-component spinor. As

is of

fermionic nature, it is described by anti-commuting Grassmann numbers [

]. The

matrices are complex 4

4 matrices and satisfy the Dirac algebra, given by the

anticommutation relations

{

, γ

}

≡ γ

+ γ

= 2η

µν

. (2.1.6)

2.1 Gauge theories in the continuum

An explicit representation of the γ

matrices is given by the Weyl or chiral basis





, γ



−σ



, γ



−



, (2.1.7)

where

refers to the 2 × 2 identity matrix, the σ

refer to the Pauli matrices





, σ



−i



, σ



−1



(2.1.8)

and we introduced the chirality matrix

≡ iγ

. We note that

anticom-

mutes with all the other γ

matrices.

The Dirac equation implies the Klein-Gordon equation



∂

+ m



Ψ = 0. (2.1.9)

This is a manifestly covariant second-order partial diﬀerential equation and ex-

presses the relativistic energy-momentum relation. Its solutions describe spinless

particles. The Dirac equation can be seen as a “square root” of the Klein-Gordon

equation, in the sense if Eq.

(2.1.5)

is multiplied by its complex conjugate, one

recovers Eq. (2.1.9).

Finally, the corresponding action of free Dirac fermions is given by



x Ψ

(

)



iγ

∂

− m



(

)

, (2.1.10)

where

Ψ ≡ Ψ

†

denotes the Dirac adjoint of

. In the presence of a gauge ﬁeld,

the partial derivatives

∂

are replaced by the covariant derivative

as discussed

in the following subsection.

2.1.2 Gauge theories

We continue by brieﬂy reviewing the formulation of the gauge sector. We discuss

both quantum electrodynamics as well as Yang-Mills theory.

The action

Quantum electrodynamics is a gauge theory with Abelian symmetry group U

(

)

The action is given by

= −





µν



, (2.1.11)

where F

µν

= ∂

− ∂

is the ﬁeld-strength tensor with vector potential A

2 Gauge theories in a nutshell

The general case of a compact, semi-simple Lie group is described by a Yang-

Mills theory [

]. Here, we restrict ourselves to the case of the special unitary groups

(

)

. The action reads

= −





µν



, (2.1.12)

where

µν

= ∂

− ∂

+ g f

abc

is the ﬁeld-strength tensor, the

denote

the gauge ﬁelds with group index

a =

, . . . , N

−

1 and

is the coupling constant.

Coupling to matter

We can couple the gauge ﬁeld to matter by replacing the partial derivatives in

Eq.

(2.1.10)

by covariant derivatives. In the case of quantum electrodynamics, this

replacement takes the form

∂

→ D

= ∂

+ ieA

, (2.1.13)

where

−e

denotes the electrical charge of the electron. Note that the ﬁeld-strength

tensor can be represented as



, D



= ieF

µν

. The action for quantum electrody-

namics then takes the form

QED





+ L



= Ψ



iγ

− m



Ψ,

= −



µν



(2.1.14)

By construction, the action

QED

is invariant under local U

(

)

transformations, for

which the inﬁnitesimal form reads

(

)

→

(

1 + iα

)

(

)

(

)

→ Ψ

(

)(

1 − iα

)

(

)

→ A

(

)

−

∂

(2.1.15)

with a scalar function α ≡ α

(

)

For a Yang-Mills theory this generalizes to

∂

→ D

= ∂

− ig A

, (2.1.16)

where



, D



= −ig F

µν

. The

denote the generators of the gauge group

(

)

, which obey commutation relations of the form



, t



= if

abc

with

2.1 Gauge theories in the continuum

structure constants f

abc

. The resulting action reads





+ L



= Ψ



iγ

− m



Ψ,

= −



µν



(2.1.17)

The inﬁnitesimal transformation law takes the form

(

)

→

(

1 + iα

)

(

)

(

)

→ Ψ

(

)(

1 − iα

)

, (2.1.18)

(

)

→ A

(

)

∂

+ f

abc

with scalar functions α

≡ α

(

)

The case of quantum chromodynamics

An important case throughout this thesis is quantum chromodynamics and there-

fore in the following we shall introduce some relevant terminology. Quantum

chromodynamics is the gauge theory of strong interactions with gauge group

(

)

. The fermions are referred to as quarks and come in three colors. We say that

the quarks are in the fundamental representation of

(

)

, while the eight gluon

ﬁelds

(

a =

, . . . ,

8) transform under the adjoint representation of the

(

)

color group. Hadrons come as color-singlet states and both quarks and gluons

have never been observed as free particles.

2.1.3 Chiral symmetry

One of the central symmetries discussed in this thesis is chiral symmetry. For the

discussion of this symmetry we follow Ref. [16] and write the Lagrangian density

= Ψ iγ

Ψ (2.1.19)

of a massless Dirac fermion. We observe the invariance of the Lagrangian density

under global chiral rotations of the form

(

)

→ e

iαγ

(

)

, Ψ

(

)

→ Ψ

(

)

iαγ

(2.1.20)

2 Gauge theories in a nutshell

and note that the presence of a mass term breaks this symmetry explicitly. We can

reformulate chiral symmetry also elegantly in the form of the anticommutation

relation

{

D, γ

}

= 0, (2.1.21)

where D = iγ

is the massless Dirac operator.

When considering

massless fermion ﬂavors, one ﬁnds a larger group of chiral

symmetry transformations of the form

(

)

→ e

iαγ

(

)

, Ψ

(

)

→ Ψ

(

)

iαγ

, (2.1.22)

(

)

→ e

iαγ

(

)

, Ψ

(

)

→ Ψ

(

)

iαγ

, (2.1.23)

where the spinors now carry an implicit ﬂavor index and the

are the generators

of the group SU

(

)

with a = 1, . . . , N

− 1.

2.1.4 Path integrals

A systematic approach to the study of quantum ﬁeld theories is the path integral

formalism [

]. It is also the starting point for the lattice approach, giving rise

to a discrete formulation suitable for numerical simulations.

To illustrate the formalism, let us consider an action

[

]



x L

[

]

of some

ﬁelds

, which we collectively refer to as

φ ≡

{

}

, and an observable

[

]

. If we

ignore some subtleties regarding the rigorous deﬁnition of the path integral for the

moment (cf. Ref. [

]), the time-ordered vacuum expectation value can be written



Dφ O

[

]

[

]

, Z =



Dφ e

[

]

. (2.1.24)

The notation



Dφ ≡





Dφ

represents an inﬁnite-dimensional integration

over all possible ﬁeld conﬁgurations of the

ﬁelds. The partition function is

denoted as Z and ensures the normalization of the expectation value.

When used for the study of gauge theories, one has to integrate over all gauge

ﬁelds

and fermion ﬁelds

and evaluate integrations of the form



DA DΨ DΨ

Due to gauge invariance, expectation values of observables are ill-deﬁned due to

over-counting of physical degrees of freedom. This can be overcome by imposing a

gauge ﬁxing condition using the Faddeev-Popov trick [

]. For more details of the

path integral method, we refer the reader to the widely available textbooks such as

Refs. [13–15].

2.2 Lattice gauge theories

One of the standard tools for studying quantum ﬁeld theories is perturbation theory.

It has proven itself to be an extremely successful method, in particular when applied

to quantum electrodynamics. As one of the major achievement of quantum physics,

the theoretical prediction of the electron’s anomalous magnetic dipole moment

agrees with experiment to more than ten signiﬁcant ﬁgures [

]. In quantum

electrodynamics, perturbative expansions take the form of a power series in the

ﬁne-structure constant

α ≈

137. As

α 

1, higher terms in the expansion are

typically strongly suppressed and one can derive accurate predictions by including

only the ﬁrst few orders.

For quantum chromodynamics, the coupling constant

at high energies or

short distances is also small. In this setting quantum chromodynamics can be

studied by perturbative methods as well. However, in the important low-energy

regime of quantum chromodynamics where bound hadron states are present, the

coupling constant

of the strong nuclear interaction is of order

(

)

and pertur-

bation theory is rendered inapplicable. Here lattice quantum chromodynamics is

the only known framework for calculating physical observables, such as hadron

masses, from ﬁrst principles. After its proposal by Wilson in Refs. [

], the lattice

approach has become one of the standard tools for non-perturbative studies of the

strong nuclear interaction. In order to arrive at a lattice formulation of a quantum

ﬁeld theory, one replaces continuous Euclidean space-time by a discrete lattice.

The discretization of the ﬁelds, the operators and the path integral then makes the

use of numerical methods possible.

In this section, we begin by reviewing the Wick rotation in Subsec. 2.2.1, followed

by the introduction of naïve lattice fermions in Subsec. 2.2.2. We then discuss

the problem of fermion doubling in Subsec. 2.2.3 and try to get an understanding

of the implications of the Nielsen-Ninomiya No-Go theorem in Subsec. 2.2.4. In

Subsec. 2.2.5, we discuss the coupling to gauge theories before we introduce Wilson

fermions in Subsec. 2.2.6, staggered fermions in Subsec. 2.2.7 and staggered Wilson

fermions in Subsec. 2.2.8. Finally in Subsec. 2.2.9, we discuss the discretization of

the gauge action on the lattice.

2 Gauge theories in a nutshell

2.2.1 Wick rotation

In Minkowski space-time the complex factor

exp

(

)

appears in the path integral,

forbidding a probabilistic interpretation of expectation values. In order for Monte

Carlo techniques to be applicable, we move to Euclidean space-time. To that end

we analytically continue the time-component of four-vectors to purely imaginary

values. Instead of the four-vector



, x



, we consider

(

x, x

)

with

≡ ix

. After this so-called Wick rotation [

], we ﬁnd a weight factor

exp

(

−S

)

with Euclidean action S in the path integral.

In Euclidean space-time the Lorentz group reduces to the four-dimensional

rotation group. For convenience we replace the Dirac algebra as stated in Eq.

(2.1.6)



, γ



= 2δ

µ, ν

. (2.2.1)

An explicit representation of the γ

matrices in the chiral basis is given by



iσ

−iσ



, γ





, γ



−



(2.2.2)

with the Euclidean chirality matrix

≡ γ

. In Euclidean space-time the

action of a gauge theory as discussed in Subsec. 2.1.2 takes the form

S =





+ L



, L



+ m



Ψ, (2.2.3)

where

with

being the respective ﬁeld-strength tensor of the theory.

Note that in Euclidean space-time the lowering or raising of indices has no eﬀect,

so we consequently write lowered indices.

2.2.2 Naïve lattice fermions

After having moved to Euclidean time, we can now begin with the discretization

of the path integral. We replace the continuous space-time domain by a discrete

hypercubic lattice

Ω = (a

Z) × (a

Z) (2.2.4)

with lattice spacings

in the spatial directions and

in the temporal

direction. In numerical simulations one considers a hypercubic subdomain

Λ ⊆ Ω

2.2 Lattice gauge theories

of the form

Λ =

























0 ≤ n

< N

0 ≤ n

< N

0 ≤ n

< N

0 ≤ n

< N











(2.2.5)

with

and

being the number of slices in each respective dimension.

On this ﬁnite domain one usually imposes (anti-)periodic boundary conditions for

the ﬁelds. To simplify our discussion, we restrict ourselves to the most common

special case of an isotropic lattice

a ≡ a

= a

, (2.2.6)

where the lattice spacing is the same along all axes.

Having introduced the lattice domain, we discretize the derivative in Eq.

(2.2.3)

which can be done in a straightforward manner by using the antihermitian sym-

metric diﬀerence operator

∂

(

)

→ ∇

(

)

≡

[

(

x + a ˆµ

)

− Ψ

(

x − a ˆµ

)]

, (2.2.7)

where

ˆµ

refers to a unit vector in

-direction. This discretization has the correct

continuum limit

lim

a→0

∇

(

)

= ∂

(

)

, (2.2.8)

namely it reproduces the derivative operator for a → 0.

In the action, the space-time integration is replaced by a sum over the lattice

sites, i.e.



x → a



x ∈Λ

. The resulting action reads

= ΨD

Ψ ≡ a



x, y ∈Λ

(

)

(

x, y

)

(

)

(2.2.9)

with the lattice Dirac operator

= γ

∇

+ m

. (2.2.10)

The operator density

(

x, y

)

is here deﬁned as

DΨ

(

)



(

x, y

)

(

)

for an

operator D, which yields in our case

(

x, y

)



y,x+a ˆµ

− δ

y,x−a ˆµ



+ m

x, y

. (2.2.11)

We note that the Dirac operator respects chiral symmetry in the sense that



, γ



2 Gauge theories in a nutshell

0, where D

denotes the massless naïve lattice Dirac operator.

In our discrete setting the path integral measure now takes the form



DΨ DΨ →



x, µ

dΨ

(

)



y, ν

dΨ

(

)

(2.2.12)

and expectation values translate to





x, µ

dΨ

(

)



y, ν

dΨ

(

)



Ψ, Ψ



−S



Ψ, Ψ



, (2.2.13)

Z =





x, µ

dΨ

(

)



y, ν

dΨ

(

)

−S



Ψ, Ψ



. (2.2.14)

We emphasize that we discretized here in the simplest possible way. As the name

suggests, naïve lattice fermions are not suitable for actual simulations as we explain

in the following subsection.

2.2.3 Fermion doubling

To understand a fundamental problem of the naïve discretization, let us compute

the free-ﬁeld propagator by inverting the lattice Dirac operator in Eq. (2.2.11). For

this well-known derivation we follow Ref. [

]. We begin by Fourier transforming

and writing the result as

(

p, q

)

= δ

(

p − q

)

(

)

with

(

)

sin





+ m

. (2.2.15)

Here the hat indicates a Fourier transformed expression. We can invert

(

)

ﬁnd

−1

(

)

− iγ

sin







sin

(

)

. (2.2.16)

Specializing it to the massless case, Eq. (2.2.16) reduces to

−1

(

)



−iγ

sin







sin

(

)

. (2.2.17)

We can verify that Eq.

(2.2.17)

has the correct continuum limit for ﬁxed values

. The resulting propagator

−iγ

of a massless fermion has a pole at

p =

(

0, 0, 0, 0

)

only. On the other hand, we observe that the lattice propagator

in Eq.

(2.2.16)

has a pole whenever all components take up values

∈

{

0, π/a

}

In addition to the physical pole, we ﬁnd additional 15 unphysical poles in the

2.2 Lattice gauge theories

corners of the Brillouin zone, among them e.g.

p =

(

π/a, 0, 0, 0

)

and

p =

(

π/a, π/a, π/a, π/a

)

The appearance of these spurious states on the lattice is the notorious fermion

doubler problem. The presence of these doubler states is related to the use of

the symmetric lattice derivative in Eq.

(2.2.11)

. However, the use of left or right

derivatives would give rise to non-covariant contributions, which would render

the theory non-renormalizable [29].

2.2.4 The Nielsen-Ninomiya No-Go theorem

A more complete understanding of the phenomenon of fermion doublers and their

connection to chiral symmetry is given by the Nielsen-Ninomiya No-Go theorem

[

–

]. It states that there can be no net chirality under a set of relatively general

conditions, namely the lattice Hamiltonian of a fermion being local, quadratic in

the ﬁelds and invariant under both lattice translations as well as under a change of

the phase of the ﬁelds [

]. In this case the number of left-handed and right-handed

states must be equal.

This theorem explains the diﬃculties in constructing fermion formulations with

all the properties one would like to have. Up to this day, no theoretically sound

lattice fermion formulation is known which satisﬁes all of the following nontrivial

conditions:

1. Absence of fermion doublers;

2. Respecting chiral symmetry;

3. Being computationally eﬃcient.

In a realistic setting (e.g. dynamical four-dimensional quantum chromodynamics)

one can generally satisfy at most two of the three conditions and we evaluate the

novel fermion discretizations discussed in this thesis with respect to these points.

2.2.5 Coupling to gauge fields

Up to this point we only considered naïve lattice fermions in the free-ﬁeld case. If

we seek to couple fermions to a gauge ﬁeld, we have to include a suitable interaction

term in the fermion action.

A peculiarity of the lattice formulation is that one introduces the gauge ﬁelds as

elements of the gauge group

, rather than as elements of the Lie algebra

like in

2 Gauge theories in a nutshell

the continuum. In the following, we restrict ourselves to the case of the (special)

unitary groups, i.e. for all U ∈ G the deﬁning property U

†

= U

−1

holds.

Let us now introduce (oriented) gauge links

(

)

∈ G

for

µ =

, . . . ,

4 and

x ∈ Λ

. We include interactions with the gauge ﬁeld by incorporating the gauge

links in the ﬁnite diﬀerence operator via

∇

(

)

≡



(

)

(

x + a ˆµ

)

−U

†

(

x − a ˆµ

)

(

x − a ˆµ

)



, (2.2.18)

resulting in

(

x, y

)



(

)

y,x+a ˆµ

−U

†

(

x − a ˆµ

)

y,x−a ˆµ



+ m

x, y

. (2.2.19)

Note that the fermionic degrees of freedom live on the lattice sites, while the gauge

degrees of freedoms are located on the links connecting neighboring sites. If we

now choose Ω

(

)

∈ G and do a gauge transformation of the form

(

)

→ Ω

(

)

(

)

(

)

→ Ψ

(

)

Ω

†

(

)

(

)

→ Ω

(

)

(

)

Ω

†

(

x + a ˆµ

)

(2.2.20)

we ﬁnd that the action

= ΨD

is invariant with respect to the symmetry

group

. This transformation is the lattice equivalent of the continuum gauge

transformations as described in Eqs. (2.1.15) and (2.1.18).

2.2.6 Wilson fermions

Let us now return to the problem of fermion doublers. Wilson’s original proposal

[5, 6] for a lattice Dirac operator can be written in the form

(

)

= γ

∇

+ m

+ W

, W

= −

∆ (2.2.21)

and is the result from adding the so-called Wilson term

to the naïve lattice

Dirac operator. The parameter

r ∈

(

0, 1

]

denotes the Wilson parameter and

∆

the covariant lattice Laplacian. The corresponding action reads

= ΨD

. In

terms of the parallel transporters

(

)

= U

(

)

(

x + a ˆµ

)

, (2.2.22)

†

(

)

= U

†

(

x − a ˆµ

)

(

x − a ˆµ

)

= T

−1

(

)

, (2.2.23)

2.2 Lattice gauge theories

we can write all terms concisely as

∇



−T

†



, ∆ =





−



, C



+ T

†



. (2.2.24)

We note that

†

= γ

and therefore

det D

∈ R

. For the important case of two

degenerate fermion ﬂavors, we can construct an action in terms of the operator

†

with det



†



≥ 0.

Due to the Wilson term, the free-ﬁeld propagator in momentum space changes

from Eq. (2.2.16) to

−1

(

)

(

)

− iγ

sin





(

)



sin

(

)

(2.2.25)

with

(

)

= m

r a

−1



sin



a/2



, see e.g. Ref. [

]. In the corners of the

Brillouin zone the mass term is eﬀectively modiﬁed to

r n/a

, where

is the

number of components of the lattice momentum

with value

π/a

. While the

physical mode is unaﬀected, the fermion doublers acquire a mass of



−1



. In

total we ﬁnd four doubler branches, where the

branch has a mass of

r k /a

and contains





modes. In the continuum limit the number of ﬂavors is then

reduced from sixteen to a single physical ﬂavor.

A shortcoming of Wilson fermions is that even for

0 chiral symmetry is bro-

ken, i.e.

{

, γ

}

0. Nevertheless, Wilson fermions and their improved versions

remain some of the most popular choices for lattice theoretical calculations due to

their conceptual simplicity.

2.2.7 Staggered fermions

An alternative approach to the fermion doubling problem are staggered fermions,

also known as Kogut-Susskind fermions [

–

]. Here one lifts an exact four-fold

degeneracy of naïve lattice fermions. As a result, the number of fermion species is

reduced from sixteen to four (in four space-time dimensions).

Let us brieﬂy review the construction of staggered fermion, where we follow

Ref. [19]. Our starting point is the naïve fermion action

= a



(

)



[

(

x + a ˆµ

)

− Ψ

(

x − a ˆµ

)]

+ m

(

)



(2.2.26)

2 Gauge theories in a nutshell

in the free-ﬁeld case. We can do a local change of variables of the form

(

)

→ Γ

(

)

(

)

, Ψ

(

)

→ χ

(

)

†

(

)

, (2.2.27)

where Γ

(

)

is a unitary matrix. By choosing Γ

(

)

so that

†

(

)

(

x + a ˆµ

)

= η

(

)

, (2.2.28)

where

(

)

is a scalar function, we can achieve a “spin diagonalization”. That this

is possible can be veriﬁed explicitly for the choice

(

)



, η

(

)

(

−1

)



µ−1

ν=0

(2.2.29)

with η

(

)

= 1. The action now takes the form

= a



ν=1



(

)



(

)[

(

x + a ˆµ

)

− χ

(

x − a ˆµ

)]

+ m

(

)



(2.2.30)

where we wrote the spinor index

explicitly. After this change of variables, we can

see that there are four identical copies in the sum over

. We can lift this degeneracy

manually by omitting three of them, giving rise to the staggered fermion action

= a



(

)



(

)[

(

x + a ˆµ

)

− χ

(

x − a ˆµ

)]

+ m

(

)



. (2.2.31)

We emphasize that

(

)

is now a one-component spinor and all that remains

from the

matrices is the staggered phase factor

(

)

. By gauging the derivative

operators, we can couple staggered fermions to a gauge ﬁeld as discussed in Subsec.

2.2.5. In summary, the staggered action and the staggered Dirac operator read

= χD

χ, D

= η

∇

+ m

, (2.2.32)

where we again make use of the matrix-vector-notation. We note that

†

= D



where



(

)

≡

(

−1

)



, (2.2.33)

and det D

≥ 0 as all eigenvalues of D

come in complex conjugate pairs.

As the action in Eq.

(2.2.31)

describes four fermion species, a natural problem is

the one of ﬂavor identiﬁcation. The two common approaches are in coordinate

space [

] or in momentum space [

–

]. We note that both approaches

2.2 Lattice gauge theories

are identical in the continuum limit [

] and quickly review the schemes in the

following.

Flavor identification in coordinate space

We begin with the identiﬁcation of ﬂavors in coordinate space, following Refs. [

]. To this end let us relabel the ﬁelds as

(

)

≡ χ

(

2x + a ρ

)

, where the multi-

index takes values

ρ ∈

{

0, 1

}

. We remark that the ﬁeld

(

)

eﬀectively lives on a

lattice with spacing

and that

(

2x + a ρ

)

= η

(

a ρ

)

. We then deﬁne the

matrix-valued fermion ﬁelds

(

)



(

a ρ

)

(

)

(

)



(

)



(

a ρ

)

(2.2.34)

This relation can be inverted to give

(

2x + a ρ

)

= 2 tr



†

(

a ρ

)

(

)



(

2x + a ρ

)

= 2 tr



(

)

(

a ρ

)



(2.2.35)

Reformulating the free staggered fermion action using these new ﬁelds, we eventu-

ally ﬁnd

= a



f ,x

(

)



∇

(

)

−





f g

∆

(

)

+ m

(

)



, (2.2.36)

where we introduced a ﬂavor index

f =

, . . . ,

4 via

(

)

≡ Ψ

µf

(

)

. Furthermore,

all ﬁnite diﬀerence operators are with respect to the lattice spacing

of the blocked

lattice and we deﬁne

≡ γ

. While the ﬁrst and third term in Eq.

(2.2.36)

is the

usual kinetic and mass term of the four fermion species

, the second term has

the form of a Wilson term with an additional nontrivial spin-ﬂavor structure.

In contrast to Wilson’s construction, staggered fermions partially preserve chiral

symmetry in the massless case. For

0, the staggered fermion action has a

(

)

× U

(

)

symmetry, given by the transformations

(

)

→ e

iα

(

)

, Ψ

(

)

→ Ψ

(

)

−iα

, (2.2.37)

(

)

→ e

i βΘ

(

)

, Ψ

(

)

→ Ψ

(

)

i βΘ

. (2.2.38)

2 Gauge theories in a nutshell

Here

≡ γ

⊗ ξ

is the generator of the remnant chiral symmetry, which can be

also expressed as



, 



0 with

being the massless staggered Dirac operator.

Flavor identification in momentum space

Another method of ﬂavor identiﬁcation is carried out in momentum space. We

follow Ref. [

] in the discussion and refer the reader to this reference for a more

complete discussion.

First note that we can write the staggered phase as

(

)

= e

iδ

·x/a

with δ

= π for ν < µ and δ

= 0 else. In momentum space we can write



−π

(

2π

)

(

2π

)

◦

(

)

(

q, p

)

◦

(

)

, (2.2.39)

where the lattice momenta

and

are in dimensionless lattice units,

◦

is 2

periodic, the Fourier transformed Dirac operator reads

(

q, p

)

(

2π

)



i sin





(

p − q + δ

)

+ m

(

p − q

)



, (2.2.40)

and

refers to the periodic Kronecker

. If we deﬁne multi-indices

and

, which

each sum over all elements in

{

0, 1

}

, we can rewrite the staggered action as



A,B

π/2



−π/2

(

2π

)

(

2π

)

(

)

A,B

(

q, p

)

(

)

(2.2.41)

with

A,B

(

q, p

)

(

2π

)



iπB

i sin





(

p − q + πB − πA + δ

)

(

p − q + πB − πA

)



. (2.2.42)

The ﬁelds are deﬁned as

(

)

≡ χ

◦

(

p + πA

)

and

(

)

≡ χ

◦

(

p + πB

)

. Now

note the relations

(

p − q + πB − πA

)

= δ

A,B

(

p − q

)

, (2.2.43)

(

p − q + πB − πA + δ

)

= ϕ

A,B

(

p − q

)

, (2.2.44)

2.2 Lattice gauge theories

where

A,B

≡





(

πB−πA+δ

)

+ 1



. (2.2.45)

If we now deﬁne

A,B

≡ exp



iπB



A,B

, we can show that they satisfy the Dirac

algebra

{

, Γ

}

µ, ν

and that the

are unitary equivalent to

⊗

spin

⊗

ﬂavor language. By using the above relations and adopting the spin

⊗

ﬂavor

notation, Eq. (2.2.41) takes the form

π/2



−π/2

(

2π

)

(

)





⊗



i sin





+ m

(

⊗

)



(

)

, (2.2.46)

where

now has an implicit spin and ﬂavor index. The action in this form is

invariant under the full U

(

)

⊗

(

)

chiral symmetry group, but this comes at the

price of a non-local action in position space [19].

2.2.8 Staggered Wilson fermions

The construction of staggered Wilson fermions was originally proposed by Adams

[

] and later extended by Hoelbling [

]. Starting from the staggered fermion

action a suitable ﬂavored mass term is added, reducing the number of doublers and

making them suitable kernel operators for the overlap and domain wall fermion

construction. Due to the central role of staggered Wilson fermions in this thesis, we

dedicate Chapter 3 to the discussion of their derivation, properties and symmetries.

2.2.9 Gauge fields

In addition to the fermionic ﬁelds, we need to formulate gauge ﬁelds on the lattice.

To this end we have to discretize the continuum gauge action given in Subsec. 2.1.2.

We require the lattice gauge action to be formulated in terms of the link variables

and to respect gauge invariance. The simplest gauge invariant objects we can form

with gauge links are closed loops. This leads us to the introduction of the so-called

plaquette

µν

(

)

≡ U

(

)

(

x + a ˆµ

)

†

(

x + a ˆν

)

†

(

)

, (2.2.47)

which is a minimal closed loop in the

-plane. We can then formulate a lattice

gauge action in terms of plaquettes only, ensuring gauge invariance by construc-

tion.

2 Gauge theories in a nutshell

In the case of an Abelian U

(

)

gauge theory, let us consider

= β



µ<ν



1 −U

µν

(

)



, (2.2.48)

where

β =

is the inverse coupling. One can verify that in the limit of zero lattice

spacing and by rescaling

→ A

, one recovers the well-known continuum

expression for the action.

Although in the context of non-Abelian gauge theories we are primarily interested

in quantum chromodynamics, it makes sense to discuss the general case of the

symmetry group

(

)

for

≥

2. An important diﬀerence compared to the

Abelian case is that the ordering in Eq.

(2.2.47)

is now important as link variables

do not commute. The lattice action now generalizes to



µ<ν



−U

µν

(

)



, (2.2.49)

where

β =

and

is the

×N

identity matrix. One can again easily verify

the correct continuum limit of the lattice discretization.

While the Wilson gauge action [

] in Eq.

(2.2.49)

is the simplest case of a gauge

action on the lattice, there are also improved actions in use. For examples we refer

the reader here to the Iwasaki [

] and DBW2 [

] actions, see also Ref. [

2.3 Numerical simulations

One of the major applications of the lattice approach to quantum ﬁeld theory is

the simulation on a computer using Monte Carlo methods. Instead of giving a

detailed discussion of how a continuum limit is taken and physical observables

can be extracted, we give a short conceptual overview of the Monte Carlo method

in the context of lattice gauge theory and refer the reader to the many excellent

textbooks in the ﬁeld, such as Refs. [

–

]. For a more general overview of the

Monte Carlo method, see e.g. Ref. [52].

2.3.1 Monte Carlo method

Assume we want to determine the expectation value of an observable

, e.g. in

order to extract a hadron mass, in the setting of a lattice gauge theory. We then

2.3 Numerical simulations

have to evaluate the expression



DΨDΨDU O



Ψ, Ψ, U



−S



Ψ, Ψ,U



, (2.3.1)

Z =



DΨDΨDU e

−S



Ψ, Ψ,U



, (2.3.2)

where the combined fermion and gauge action reads



Ψ, Ψ, U



= S



Ψ, Ψ, U



+ S

[

]

(2.3.3)

and the generalization to several fermion species is straightforward. Assuming

the fermionic degrees of freedom appear quadratically, we can integrate them out

exactly. We arrive at an integrand, which only depends on the gauge ﬁeld and reads



DU O

[

]

det D

[

]

−S

[

]

, (2.3.4)

Z =



DU det D

[

]

−S

[

]

. (2.3.5)

We note that the integral over the spinors

is of the Grassmann / Berezin

kind [

] and gives rise to the fermion determinant

det D

[

]

with the lattice Dirac

operator

. The operator

[

]

follows from



Ψ, Ψ, U



after integrating out

the fermions in the path integral with the help of Wick’s theorem [

]. We note that

in general O

[

]

contains propagators of the form D

−1

[

]

To give an explicit example following Ref. [

], consider two Dirac fermion species

, Ψ

and let

O = Ψ

(

)

Γ Ψ

(

)

(

)

Γ Ψ

(

)

(2.3.6)

be an iso-triplet operator, where

is a monomial of

matrices. After integrating

out both the Ψ

and Ψ

ﬁelds, one ﬁnds

= −tr



Γ D

−1

(

x, y

)

Γ D

−1

(

y, x

)



, (2.3.7)

where

and

are the lattice Dirac operators of the respective fermion species.

As the resulting integral is of extremely high dimension, one makes use of Monte

Carlo methods for the numerical evaluation. To this end one interprets the factor

det D

[

]

−S

[

]

, (2.3.8)

the so-called Gibbs-measure, as a probability measure. One ﬁrst generates a

2 Gauge theories in a nutshell

Markov chain of gauge conﬁgurations

with respect to this probability distribu-

tion. Under certain conditions we can then replace the path integral expectation

value by

= lim

N →∞



i =1

[

]

, (2.3.9)

where we take an ensemble average.

The computationally most expensive part is the generation of the gauge ensem-

bles according to the probability distribution in Eq.

(2.3.8)

due to the presence of the

fermion determinant

det D

[

]

. Moreover, as one approaches the continuum limit

and one gets closer to the physical point, the computational costs rapidly increase,

see e.g. Refs. [

]. While in the early days of lattice quantum chromodynamics

numerical simulations where done in the so-called quenched approximation, i.e.

one does the replacement

det D

[

]

→

1, dynamical simulations are nowadays

state of the art.

The high computational costs of these Monte Carlo simulations explain why

research in novel lattice fermion formulations remains a very active ﬁeld, see e.g.

Refs. [56, 57] for an overview.



C 

S W 

The central topic of this thesis are staggered Wilson fermions and their related

formulations, namely staggered overlap and staggered domain wall fermions. For

this reason, we dedicate this chapter to the discussion of ﬂavored mass terms and

the construction of staggered Wilson fermions.

The staggered Wilson fermion formulation ﬁrst arose in investigations regarding

the index theorem on the lattice using staggered fermions [

]. The lattice Dirac

operator which emerged from these investigations [

] shares properties of both

staggered and Wilson fermions. Later Hoelbling expanded upon this idea and

proposed a related fermion formulation [

]. In Ref. [

], we went one step further

and introduced generalizations of these constructions for arbitrary ﬂavor splittings

in arbitrary even dimensions.

3.1 Introduction

To understand the idea behind staggered Wilson fermions, we recall that there are

two traditional approaches to deal with the problem of fermion doublers. Wilson’s

original approach is to add the Wilson term as discussed in Subsec. 2.2.6, which

is a symmetric covariant discretization of the Laplacian. The eﬀect of this term

is that the spurious fermion states acquire a mass of



−1



and, thus, decouple

in the continuum limit. The other approach is the one of staggered fermions as

introduced in Subsec. 2.2.7. Here one lifts an exact degeneracy of naïve fermions

using a “spin diagonalization”. As a result, not all fermion doublers are removed,

but the number of species is reduced from sixteen to four. We note that in practical

simulations one can further reduce the number of ﬂavors to

by taking an appro-

priate root of the fermion determinant. Explicitly, one can make a replacement of

the form

det D

→

(

det D

)

. (3.1.1)

3 Staggered Wilson fermions

Due to the lack of a formal proof of correctness, the method remained controversial

for a long time and one can ﬁnd arguments in favor [

–

] and against [

–

]

this approach in the literature, see also Refs. [67, 68] for an overview.

Staggered Wilson fermions combine both approaches by taking the staggered

fermion action and adding a suitable “staggered Wilson term” to further reduce the

number of doublers. This staggered Wilson term turns out to be a combination of

the ﬂavored mass terms as discussed in detail in the classical paper by Golterman

and Smit [

]. Due to the presence of this term, staggered Wilson fermions have

technical properties similar to Wilson fermions. By using them as a kernel opera-

tor, they allow the construction of staggered overlap and staggered domain wall

fermions. Usual staggered fermions, on the other hand, do not deﬁne a suitable

kernel operator. This is because these constructions rely on the property

which does not hold for its staggered equivalent as we discuss in more detail in

Sec. 6.2.

In order to understand the construction of the staggered Wilson term and its

variants, we begin by discussing symmetries of the staggered fermion action in Sec.

3.2 and properties of ﬂavored mass terms in Sec. 3.3. In Sec. 3.4, we then review

Adams’ proposal, followed by Hoelbling’s construction in Sec. 3.5. We end this

chapter with the discussion of our generalized mass terms in Sec. 3.6.

3.2 Symmetries of staggered fermions

In the following we discuss the symmetries of the massless staggered fermion

action

S =



x, µ

(

)



¯χ

(

)

(

)

(

x + a ˆµ

)

− ¯χ

(

)

†

(

x − a ˆµ

)

(

x − a ˆµ

)





x, µ

(

)



¯χ

(

)

(

)

(

x + a ˆµ

)

− ¯χ

(

x + a ˆµ

)

†

(

)

(

)



. (3.2.1)

The action is invariant under the transformations listed below, where we follow

the discussion and notation of Ref. [

] (cf. Ref. [

]). We note that in order for the

path integral measure to be invariant under all these transformations, we consider

an even number of lattice sites and (anti-)periodic boundary conditions.

3.2 Symmetries of staggered fermions

Shift invariance. The transformation is given by

(

)

→ ζ

(

)

(

x + a ˆρ

)

¯χ

(

)

→ ζ

(

)

¯χ

(

x + a ˆρ

)

(

)

→ U

(

x + a ˆρ

)

(3.2.2)

with

(

)

(

−1

)



ν=µ+1

(3.2.3)

and ζ

(

)

= 1.

Rotational invariance. The transformation is given by

(

)

→ S



−1





−1



¯χ

(

)

→ S



−1



¯χ



−1



(

x, y

)

→ U



−1

x, R

−1



(3.2.4)

where

R ≡ R

ρσ

is the rotation

→ x

→ −x

→ x

with

µ , σ, ρ

. We

also introduced



−1





1 ± η

(

)

(

)

∓ ζ

(

)

(

)

+ η

(

)

(

)

(

)

(

)



(3.2.5)

for

ρ ≶ σ

and note that



−1



(

)



−1

[

x + a ˆµ

]



= R

µν



−1



. More-

over, we make use of the compact notation

(

x, y

)











(

)

, y = x + a ˆµ,

†

(

)

, x = y + a ˆµ,

(3.2.6)

for the gauge links.

Axis reversal. The transformation is given by

(

)

→

(

−1

)





¯χ

(

)

→

(

−1

)

¯χ





(

x, y

)

→ U



x, I



(3.2.7)

where I

x is the reversal of the ρ-axis, i.e. x

→ −x

, x

→ x

for µ , ρ.

3 Staggered Wilson fermions

(

)

symmetry. The transformation is given by

(

)

→ e

iα

(

)

¯χ

(

)

→ ¯χ

(

)

−iα

(3.2.8)

This symmetry is associated with the conversation of charge.

(

)



symmetry. The transformation is given by

(

)

→ e

i β

(

)

(

)

¯χ

(

)

→ ¯χ

(

)

i β

(

)

(3.2.9)

with



(

)

(

−1

)



. (3.2.10)

This is the remnant chiral symmetry of staggered fermions.

Exchange symmetry. The transformation is given by

(

)

→ ¯χ

(

)

¯χ

(

)

→ χ

(

)

(

)

→ U



(

)

(3.2.11)

which interchanges fermion and anti-fermion ﬁeld.

Remarks.

For the following discussion, let us also deﬁne the parity transforma-

tion as

followed by a shift in the 4-direction. We note, that in addition to

the position-space formulations of these symmetries, one can also express them

elegantly in momentum space. For brevity we do not quote the momentum repre-

sentation here explicitly, but refer the reader to Ref. [42] for details.

3.3 Flavored mass terms

Following Refs. [

] closely, we now discuss ﬂavored mass terms. In the classical

continuum limit we can add a mass term

cont



x ΨM Ψ (3.3.1)

3.3 Flavored mass terms

to the action. We demand that

is trivial in spinor space and that the action is

rotationally invariant. The latter requires the mass term to commute with

Using an explicit spin

⊗

ﬂavor-notation, a general ﬂavored mass term now assumes

the structure

M =

⊗



+ m

µν

+ m

iξ

+ m



, (3.3.2)

where

µν

is antisymmetric,

µν

≡ iξ

and the

are a representation of the

Dirac algebra in ﬂavor space, i.e.



, ξ



µν

. Furthermore, we require

that

≥

0 in the free-ﬁeld case, so

is taken to be Hermitian and the mass

parameters are real. We note that in Eq.

(3.3.2)

we could in principle also add

similar terms multiplied by a

⊗ ξ

, but we do not consider them in the following

discussion.

Let us now introduce the symmetric shift operator

(

x, y

)





(

)

(

x − z

)

(

y − z − a ˆµ

)

+ U

†

(

)

(

y − z

)

(

x − z − a ˆµ

)



(3.3.3)

which we expect to result in a Hermitian transfer matrix. We can implement

Eq. (3.3.2) on the lattice as

= a



m χ

(

)

(

)



x, y

(

)

(

x, y

)

(

)



x, y, z

µν

(

)

(

x, y

)

(

y, z

)

(

)

−



w, x, y, z

µα βγ

(

)

(

w, x

)

(

x, y

)

(

y, z

)

(

)

−



v,w, x, y, z

α βγδ

(

)

(

v, w

)

(

w, x

)

(

x, y

)

(

y, z

)

(

)

(3.3.4)

In order to get a better understanding of the mass term in Eq.

(3.3.4)

, let us discuss

which of the symmetries in Sec. 3.2 it breaks. First we note that in this general

form the mass term is not invariant under lattice rotations. However, violations

of rotational invariance are expected to be of order

(

)

, so that the symmetry

is eventually restored in the continuum limit. For parity transformations we ﬁnd

invariance of our mass term, while for the exchange symmetry

and

change signs.

3 Staggered Wilson fermions

For numerical applications, we require a real determinant and, thus, we need

to retain



Hermiticity. Hence the ﬂavor structure of the mass term needs to be

restricted to a sum of products of an even number of

. Therefore from now on

we restrict ourselves to the case

= m

= 0, (3.3.5)

leaving us with the m, m

µν

and m

term.

In Sec. 3.4, we follow Adams’ original two-ﬂavor proposal and construct a stag-

gered Wilson term using

and

, while in Sec. 3.5 we discuss Hoelbling’s one

ﬂavor construction using m and m

µν

. We ﬁnish this chapter with generalizations

of these ﬂavored mass terms in Sec. 3.6.

3.4 Adams’ mass term

Adams proposed a particular form of the ﬂavored mass term, giving rise to the

original two-ﬂavor staggered Wilson fermion formulation introduced in Refs. [44,

45]. In the following, we discuss its construction, interpretation and properties.

3.4.1 Construction

The staggered Wilson action reads

= ¯χD

χ = ¯χ

(

+ m

+ W

)

χ (3.4.1)

with the massless staggered Dirac operator

= η

∇

. The staggered Wilson term

reads

(

− Γ

)

(3.4.2)

with Wilson-like parameter

r >

0. Following Adams’ original notation, we intro-

duced the operator

(

)

(

−1

)



(

)

, (3.4.3)

where depending on the context we also use



(

)

(

−1

)



for the sign factor.

In Eq. (3.4.2) we introduced

= η

C , η

= η

, η

(

)

(

−1

)

(

)

(

)

(3.4.4)

3.4 Adams’ mass term

and the symmetrized product of the C



µ+

+ T

µ−



/2 operators

C =

(

)

sym

≡

sym



α βγδ

. (3.4.5)

As pointed out earlier, Adams’ ﬂavored mass term in Eq.

(3.4.2)

preserves



Her-

miticity of the Dirac operator, namely

†

= D



, and is itself Hermitian. This

implies a non-negative fermion determinant

det D

≥

0 for suitable choices of the

fermion mass

and ensures the applicability of importance sampling techniques

to Monte Carlo simulations.

3.4.2 Interpretation

To interpret Adams’ staggered Wilson term, we note that in the spin

⊗

ﬂavor inter-

pretation [42] we ﬁnd

 γ

⊗ ξ

, Γ

 γ

⊗

+ O





. (3.4.6)

While

corresponds to the usual

up to discretization eﬀects, the

is a ﬂavored

in the sense that it acts nontrivially in ﬂavor space. As a result, we ﬁnd for Adams’

staggered Wilson term the spin ⊗ﬂavor structure



⊗

(

− ξ

)

+ O

(

)

. (3.4.7)

We note that

is of the form of a projector

(

1 ± ξ

)

2 in ﬂavor space and

allows for a simple interpretation. The staggered Wilson term gives a mass



−1



to the two negative ﬂavor-chirality species, similar to the eﬀect the usual Wilson

term has on the doubler modes. In the continuum limit

a →

0, the fermion

doublers become heavy and decouple. On the other hand, the two positive ﬂavor-

chirality species do not acquire additional mass contributions. Note that the minus

(3.4.2)

and Eq.

(3.4.7)

can be replaced by a plus sign, thus interchanging

the role of positive and negative ﬂavor-chirality species.

3.4.3 Properties

An important observation is that the equivalent of

, namely

, does not square

to the identity operator. While

holds exactly, we ﬁnd

+ O





This is at the heart of the problem why usual staggered fermions are not suitable

3 Staggered Wilson fermions

kernel operators for the overlap and domain wall constructions (see Sec. 6.2 for

a more in-depth discussion). However, there is another operator which squares

exactly to unity, namely

. While

has a spin

⊗

ﬂavor interpretation of

⊗ ξ

, for Adams’ construction we project out the positive ﬂavor-chirality species

in the continuum limit and ﬁnd

= Γ

+ O





 γ

⊗

+ O





(3.4.8)

on the physical species. Therefore on the subspace of these modes one can use

instead of

. This crucial insight by Adams allows the construction of staggered

overlap and staggered domain wall fermions, which we discuss in Chapter 6, 7 and

To summarize some interesting properties of two-ﬂavor staggered Wilson fer-

mions, we recall that compared to usual staggered fermions we are left with a

reduced number of tastes, namely two instead of four (in

d =

4 dimensions). We

also ﬁnd a smaller fermion matrix and expect a better condition number compared

to Wilson fermions, potentially increasing computational eﬃciency. In addition,

staggered Wilson fermions allow the construction of staggered versions of overlap

and domain wall fermions.

On the downside, we point out that the staggered Wilson term

breaks a

subset of the staggered fermion symmetries. In particular, the exact ﬂavored chiral

symmetry is broken in the massless case, i.e.



, 



0. This gives rise to an

additive mass renormalization and there is the need to ﬁne-tune the bare mass

like for Wilson fermions. Moreover, new fermionic counterterms are allowed [

but the only eﬀect is a wave function renormalization for the physical species. On

the physical species the staggered Wilson terms vanished up to order

(

)

, hence

the order





discretization error of usual staggered fermions is lost. Finally,

we note that the

(

)

vector and chiral symmetries of the two physical ﬂavors

are broken, similarly to the broken

(

)

symmetry of usual staggered fermions.

Nevertheless, the remaining symmetries, such as the ﬂavored rotation symmetries,

are still enough to ensure e.g. a degenerate triplet of pions [

]. We also note

that the staggered Wilson term is a four-hop operator, making it potentially very

susceptible to ﬂuctuations of the gauge ﬁeld [

] and diﬃcult to parallelize. We

discuss these aspects in more detail in Chapter 4.

3.5 Hoelbling’s mass term

3.4.4 Aoki phase

We conclude our discussion of the theoretical properties of staggered Wilson fermi-

ons with the investigations of the Aoki phase [

–

] carried out in Refs. [

–

]. In

these studies, the authors investigated strong-coupling lattice quantum chromo-

dynamics with staggered Wilson fermions using hopping parameter expansions

and eﬀective potential analyses. The existence of a non-vanishing pion conden-

sate in certain mass parameter ranges could be established, while massless pions

and PCAC

relations were found around a second-order phase boundary. These

results are in support of the idea that, like with Wilson fermions, staggered Wilson

fermions can be applied to lattice quantum chromodynamics by taking a chiral

limit.

3.5 Hoelbling’s mass term

One can build upon Adams’ idea and construct ﬂavored mass terms to completely

lift the degeneracy of staggered fermions. Originally proposed by Hoelbling in four

dimensions in Ref. [46], we are going to review the construction in the following.

3.5.1 Construction

A possible approach to construct single-ﬂavor staggered fermions is to start from

Adams’ operator and add an additional term, which lifts the degeneracy of species

of the same ﬂavor-chirality. A natural candidate for this is the

µν

term in Eq.

(3.3.4)

with a spin

⊗

ﬂavor interpretation of

⊗ σ

µν

. A possible candidate for a one-ﬂavor

action [46] is then given by

S = ¯χD χ ≡ ¯χ



+ m

(2 ·

+ W

+ M

µν

)



χ, (3.5.1)

where we introduced the mass term M

µν

= iη

µν

(no sum) with operators

µν

(

)

(

−1

)

ρ=µ+1

(

)

for µ < ν,

µν

= −η

νµ

for µ ≥ ν

(3.5.2)

and

µν



, C



2. Here

refers to Adams’ staggered Wilson term as dis-

cussed in Sec. 3.4 and the choice of µ , ν is arbitrary.

PCAC: Partially Conserved Axial Current

3 Staggered Wilson fermions

While the Dirac operator in Eq.

(3.5.1)

lifts the degeneracy of the staggered fer-

mion ﬂavors completely, one can consider a more symmetric combination of the

mass terms

µν

, resulting in better symmetry properties. Hoelbling’s proposal [

]

now takes the form

= ¯χD

χ ≡ ¯χ



+ m

(2 ·

+ M

)



χ (3.5.3)

with the symmetrized ﬂavored mass term

[

(

+ s

)

+ s

(

+ s

)

+ s

(

+ s

)]

√

3 (3.5.4)

and

= ±

µν

= ±

1 being arbitrary sign factors [

]. We note that in numerical

applications a commonly employed form of Eq. (3.5.4) is given by

(

+ M

− M

+ M

)

√

3, (3.5.5)

see Refs. [

]. As

already lifts the degeneracy completely by itself, there is

no need to introduce a W

term in Eq. (3.5.3).

When compared to Adams’ staggered Wilson operator

, the mass term

has a smaller symmetry group. One can verify the invariance [

] under diagonal

shifts given by

x → x ±

1 ±

2 ±

3 ±

, shifted axis reversals given by

→ −x

+ a ˆµ

and double rotations given by

x → R

α β

γδ

, where

and

are mutually

distinct.

3.5.2 Interpretation

To understand the eﬀect of Hoelbling’s ﬂavored mass term, we note [46] that



⊗ ξ

+ O

(

)

= diag

(

4, 0, 2, 2

)

or diag

(

2, 2, 4, 0

)

(3.5.6)

in the spin

⊗

ﬂavor interpretation. One can now see that in the presence of the

mass term M

three ﬂavors are lifted to two distinct doublers points, namely two

ﬂavors to 2

r /a

and one to 4

r /a

. The remaining physical ﬂavor remains massless

under the action of the mass term, while the doublers decouple in the continuum

limit. Similar to the two ﬂavor case, the staggered overlap and staggered domain

wall constructions can be carried out for Hoelbling’s operator.

3.6 Generalized mass terms

3.5.3 Properties

The Dirac operator

breaks several symmetries of the staggered fermion action.

Among them is rotational symmetry, where only a residual subgroup survives. It

turns out that there are no additional fermionic counterterms in the action, but new

gluonic counterterms appear as pointed out by Sharpe [

]. They arise from the

fermion loop contributions to the gluonic two-, three- and four-point functions, cf.

Ref. [

]. This means that in dynamical simulations one has to include and ﬁne-tune

these terms, severely limiting the practical applicability of single ﬂavor staggered

fermions.

For this reason, starting from Chapter 4, we restrict ourselves to the case of

Adams’ original two-ﬂavor staggered Wilson kernel.

3.6 Generalized mass terms

In the following, we want to further generalize the ﬂavored mass terms discussed

in Sec. 3.4 and Sec. 3.5 to allow for arbitrary mass splittings in arbitrary even

dimensions. In this section, we follow our discussion

given in Ref. [

] closely,

where we previously presented the following results.

3.6.1 The four-dimensional case

Up to discretization terms, the staggered Wilson term is a mass term which is

trivial in spin-space, but splits the diﬀerent staggered ﬂavors. We require that

the determinant of the lattice fermion Dirac operator is real, to allow the use of

importance sampling techniques in Monte Carlo simulations. Adopting a more

convenient notation for the following discussion, the original proposal by Adams

[44, 45] can be written as

(

+ Γ

1234

)

. (3.6.1)

Here r > 0 is the Wilson-like parameter and we deﬁne the operators

1234

(

)

(

−1

)



(

)

, (3.6.2)

Discussion based with permission on C. Hoelbling and C. Zielinski, “Spectral properties and

chiral symmetry violations of (staggered) domain wall fermions in the Schwinger model,” Phys.

Physical Society.

3 Staggered Wilson fermions

1234

= η

(

)

sym

. (3.6.3)

We recall that this term has a spin

⊗

ﬂavor interpretation of the form of Eq.

(3.4.7)

and splits the four ﬂavors of staggered fermions into two pairs with opposite ﬂavor

chirality, that is, the eigenbasis of

. For the following discussion, the notation

A ∼ B means that A has the spin ⊗ﬂavor interpretation B up to proportionality.

It is also possible to split the ﬂavors with respect to the eigenbasis of other ele-

ments of the Dirac algebra in ﬂavor space [

]. We restrict the structure of the

mass term to a sum of products of an even number of

matrices, so that



Her-

miticity and the reality of the fermion determinant is retained. In four dimensions

we can, thus, reduce the number of staggered fermion ﬂavors to one by e.g.



2 ·

+ W



, (3.6.4)

µν

= i Γ

µν

, (3.6.5)

where the operators Γ

µν

and C

µν

are deﬁned as

µν

(

)

= ε

µν

(

−1

)

(

)

(

)

, (3.6.6)

µν

= η



+ C



(

no sum

)

. (3.6.7)

Here

···µ

is the totally antisymmetric Levi-Civita symbol. In order to interpret

the mass term deﬁned in Eq. (3.6.5), we note that

µν

∼ γ

⊗ ξ

, C

µν

∼ γ

⊗

,  ∼ γ

⊗ ξ

, (3.6.8)

up to discretization terms. Therefore, we ﬁnd that

µν

∼

⊗ σ

µν

+ O

(

)

(3.6.9)

with

µν

= iξ

. We now see that the number of physical ﬂavors is reduced by

to one, i.e. all but a single ﬂavor acquire a mass of O



−1



3.6 Generalized mass terms

3.6.2 The d -dimensional case

We can now generalize our constructions to an arbitrary even number of dimen-

sions d. Here we write a single ﬂavor mass term as

d/2



k =1



+ W

(

2k−1

)(

)



, (3.6.10)

so that Eq.

(3.6.4)

follows after the specialization to

d =

4. We can generalize this

mass term even further by introducing

···µ

= i

···µ

, (3.6.11)

for an arbitrary n ≤ d/2, where

···µ

(

)

= ε

···µ

(

−1

)



i =1

(

)

, (3.6.12)

···µ

= η

···η



···C



sym

. (3.6.13)

The spin ⊗ ﬂavor interpretations of these operators read

···µ

∼



···γ



⊗



··· ξ



, (3.6.14)

···µ

∼



···γ



⊗

, (3.6.15)

···µ

∼

⊗



··· ξ



, (3.6.16)

up to discretization terms. These new mass terms are  Hermitian as well, that is

†

=  M , M ≡W

···µ

, (3.6.17)

where



(

)

(

−1

)



µ=1

. (3.6.18)

This allows us to introduce a very general mass term of the form

d/2



n=1





+ W



(3.6.19)

with generalized Wilson-parameters

≥

0. Here the sum is over all multi-indices

(

, . . . , µ

)

with 1

≤ µ

≤ d

for all

with 1

≤ i ≤

. We remark, that in

Eq.

(3.6.19)

not all of the possible combinations of mass terms are useful in practical

applications. To reproduce Adams’ staggered Wilson term in

d =

4 dimensions as

3 Staggered Wilson fermions

given in Eq. (3.6.1), we set r

1234

= r > 0 and r

= 0 otherwise.

In Chapter 8 we deal with the

d =

2 case. We note that here the deﬁnition is

essentially unique and the mass term takes the form



+ W



, (3.6.20)

which reduces the number of staggered ﬂavors from two to one.

We note that all possible

terms, like the Wilson term

, break chiral sym-

metry. Moreover, if too many of the symmetries of staggered fermions are broken,

some mass terms may give rise to additional counterterms, cf. Ref. [69].



C 

C 

Staggered Wilson fermions and usual Wilson fermions share several technical

properties. Both are constructed by adding a momentum-dependent term to a

fermion action suﬀering from fermion doubling. This additional term breaks chiral

symmetry explicitly, resulting in an additive mass renormalization and the need

for ﬁne-tuning the bare mass parameter.

Besides the technical aspects, comparing the computational properties of diﬀer-

ent lattice fermion actions is important for practical applications. As the accuracy

of physical predictions in Monte Carlo simulations is limited by the available com-

putational resources, the question of computational eﬃciency is then of high

importance. In this chapter, we deal with the problem of quantifying and com-

paring the computational eﬃciency of two-ﬂavor staggered Wilson fermions with

the one of usual Wilson fermions, where some of the results

were previously pre-

sented in Refs. [

]. We try to answer the question if staggered Wilson fermions

have a computational advantage, potentially bringing down the enormous costs of

state-of-the-art numerical simulations.

4.1 Introduction

Staggered Wilson fermions, like usual staggered fermions, are formulated using

one-component spinor ﬁelds. One might, thus, hope for a better computational

eﬃciency of staggered Wilson fermions compared to Wilson fermions due to a

smaller fermion matrix and a reduced condition number. The prospect of a kernel

operator with increased computational eﬃciency is especially interesting in the

context of the overlap and domain wall fermion formulations. These formulations

Parts of the discussion based on and some ﬁgures reprinted from D. H. Adams, D. Nogradi,

A. Petrashyk, and C. Zielinski, “Computational eﬃciency of staggered Wilson fermions: A ﬁrst

look,” PoS

LATTICE2013

(2014) 353,

arXiv:1312.3265 [hep-lat]

. Copyright owned by the

authors under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives

4.0 International License (CC BY-NC-ND 4.0).

4 Computational eﬃciency

respect chiral symmetry, but come at the price of high computational costs. This

limits their practical applicability despite many attractive theoretical properties.

In this chapter, we evaluate the computational eﬃciency of Adams’ two-ﬂavor

staggered Wilson fermions. As the performance of overlap and domain wall fermi-

ons is tightly connected to the computational properties of their kernel operator,

this is also interesting with respect to our discussion of staggered overlap fermions

in Chapter 6 and 7 and staggered domain wall fermions in Chapter 8. We follow

our report given in Ref. [

] together with the updates given in Refs. [

], where we

discuss our results for the computational eﬃciency in a well-deﬁned benchmark

setting.

We compare the computational costs of inverting the Dirac matrix on a source

for staggered Wilson fermions and usual Wilson fermions in quenched lattice

quantum chromodynamics. We do this while either keeping the physical volume

or the lattice spacing ﬁxed. These inversions are typically the most time-consuming

part in realistic simulations and are, thus, expected to be a good proxy for studying

the computational performance. The resulting large and sparse linear systems

are solved using iterative methods, such as the conjugate gradient method [

]. We characterize the computational eﬃciency as the ratio of the computation

time for both fermion formulations. For this comparison to be meaningful, the

benchmarking has to be done at ﬁxed values of some physical quantity, where in

general the measured eﬃciency depends on this ﬁxed quantity. A natural candidate

for this is the pion mass

, which is easy to measure and tune with high accuracy.

This chapter is organized as follows. In Sec. 4.2, we discuss our strategy to quan-

tify the computational eﬃciency of staggered Wilson fermions and derive some

theoretical estimates. In Sec. 4.3, we present our numerical study and critically dis-

cuss our results. In Sec. 4.4, we characterize the memory bandwidth requirements

of staggered Wilson fermions by analyzing the arithmetic intensities of several

lattice fermion formulations. Finally, in Sec. 4.5 we summarize and conclude our

study.

4.2 Theoretical estimates

As we already pointed out, the computational costs of realistic simulations are

dominated by inverting the Dirac operator

D ≡ D





on a source

, namely

solving the linear system

DΨ = χ

. As these linear systems are usually very large,

4.2 Theoretical estimates

one uses iterative solvers to ﬁnd a numerical solution for

. A traditional approach

to this is the use of the conjugate gradient method for the normal equation, where

one solves the system

†

D Ψ = χ

(4.2.1)

with

≡ D

†

instead. We form these normal equations with the Hermitian and

positive (semi-)deﬁnite operator

†

to ensure the applicability of the conjugate

gradient method, see Ref. [85].

In the following, we want to derive a theoretical estimate for solving Eq.

(4.2.1)

with the conjugate gradient method. First we note that the cost can be characterized

cost =

(

#iterations

)

× (cost per iteration). (4.2.2)

For most iterative methods, such as the conjugate gradient method, the cost per

iteration is dominated by the cost of the matrix-vector multiplications. This gives

cost ∝

(

#iterations

)

(

cost mat.-vec. mult.

)

. (4.2.3)

If we now take the ratio for the computational costs of Wilson and staggered Wilson

fermions, we ﬁnd

cost

(

#iterations

)

(

#iterations

)

(

cost mat.-vec. mult.

)

(

cost mat.-vec. mult.

)

. (4.2.4)

When considering the fermion propagator, an additional factor of four should be

included on the right hand side because in the staggered Wilson case one has four

times fewer sources due to a reduced number of spinor components. To get an

estimate for the overall cost ratio of both fermion formulations, we have to estimate

both ratios on the right hand side of Eq.

(4.2.4)

. The ﬁrst ratio in general depends

on the iterative solver and the ﬁxed physical quantity, in our case the pion mass

. The second ratio depends on size, structure and sparsity of the corresponding

fermion matrices.

4.2.1 Number of iterations

We begin by further analyzing the ﬁrst ratio in Eq.

(4.2.4)

. When using the conjugate

gradient method for a speciﬁed small residual ε, we make use of the relation

#iterations ∝

√

κ log





, (4.2.5)

4 Computational eﬃciency

where

κ =

max

/λ

min

is the condition number of

†

, see e.g. Ref. [

]. Hence

one can approximate the ratio for the number of iterations by

(

#iterations

)

(

#iterations

)

≈



, (4.2.6)

when keeping

ﬁxed. To verify this, we also numerically determine this ratio. In

the free-ﬁeld case the ratio in Eq.

(4.2.6)

can be computed analytically for the same

bare mass m

for both formulations. We ﬁnd

lim

→0



√

≈ 3.6, (4.2.7)

where κ

≡ κ





refers to the free-ﬁeld condition number at bare mass m

4.2.2 Cost of matrix-vector multiplications

Let us now analyze the second ratio on the right hand side of Eq.

(4.2.4)

. We es-

timate the costs for the matrix-vector multiplications from FLOP (ﬂoating-point

operations) counts. We ﬁnd

(

cost mat.-vec. mult.

)

(

cost mat.-vec. mult.

)

≈

(

FLOPs

)

(

FLOPs

)

≈

4 ×

(

FLOPs/site

)

(

FLOPs/site

)

4 × 1392 FLOPs/site

1743 FLOPs/site

≈ 3.2. (4.2.8)

Here the factor of four appears due to the four-component spinors in Wilson’s

formulation, 1392 is the number of FLOPs per site needed for the application of

Wilson’s Dirac operator and 1743 is the number of FLOPs per site for the staggered

Wilson Dirac operator. Note that Eq.

(4.2.8)

is an estimate and besides the number

of ﬂoating point operations also other factors come into play such as the computa-

tional intensities, which we discuss in Sec. 4.4.

Alternatively one can estimate the ratio in Eq.

(4.2.8)

with the approach of

Ref. [

], which we now brieﬂy review. In practice Wilson fermions are used with the

spin-projection trick [

], eﬀectively reducing the number of spinor components

from four to two. The Wilson action couples each lattice site to 2

8 other sites,

4.3 Numerical study

0.05

0.1

0.15

0.2

−0.82 −0.8 −0.78 −0.76 −0.74 −0.72 −0.7 −0.68

Pion mass m

Quark bare mass m

Pion mass m

at β = 6, 16

x 32 lattice

Wilson, linear fit

Wilson

Stag. Wilson, linear fit

Stag. Wilson

Figure 4.3.1:

Squared pion mass

as a function of the bare quark mass

with a

linear ﬁt through the data points with 0.05 ≤ m

where the factor of two represents the eﬀective number of spinor components and

2 comes from a forward and a backward hop in each space-time direction.

Staggered Wilson fermions are formulated with one-component spinors, but the

staggered Wilson term couples each site to all 2

16 corners of a four-dimensional

hypercube. This means that the staggered Wilson action couples each lattice site

to 8 + 16 nearby sites. We can now alternatively estimate the ratio by

4 × (8 × 2)

8 + 16

≈ 2.7, (4.2.9)

where we included the above-mentioned factor of four. This estimate is smaller

than the one we gave in Eq.

(4.2.8)

as we neglected here the cost of spin decompo-

sition and reconstruction for the spin projection trick.

In summary we conservatively estimate

(

cost mat.-vec. mult.

)

(

cost mat.-vec. mult.

)

≈ 2 − 3 (4.2.10)

for the cost-ratio for the matrix-vector multiplications, in agreement with Ref. [

4 Computational eﬃciency

0.05

0.1

0.15

0.2

0.25

−0.82 −0.8 −0.78 −0.76 −0.74 −0.72 −0.7 −0.68

Pion mass m

Quark bare mass m

Pion mass m

at β = 6, 20

x 40 lattice

Wilson, linear fit

Wilson

Stag. Wilson, linear fit

Stag. Wilson

Figure 4.3.2:

Squared pion mass

as a function of the bare quark mass

with

a linear ﬁt through the data points with 0.05 ≤ m

4.3 Numerical study

We did a numerical study with quenched QCD conﬁgurations on a 16

32 lattice

β =

6, a 20

40 lattice at

β =

6 and a 20

40 lattice at

β =

136716 with 200

conﬁgurations each. The 20

40 lattices are chosen in such a way to either keep

the lattice spacing (

β =

6) or the physical volume (

β =

136716) ﬁxed compared to

the 16

32 lattice. For the simulations we used the Chroma/QDP software package

[87], which we extended by implementing staggered Wilson fermions.

While the use of preconditioning can speed up the use of Wilson fermions by

roughly a factor of two and preconditioning is in principle also possible for stag-

gered Wilson fermions, our study here deals with the unpreconditioned base case.

4.3.1 Results

In Figs. 4.3.1, 4.3.2 and 4.3.3, we ﬁnd the squared pion mass

as a function of

the bare quark mass

for the diﬀerent ensembles. The relation between the

pion mass and bare quark mass for staggered Wilson fermions was previously also

presented in Refs. [4, 88].

For the determination of the pion mass we follow the well-known standard

procedures, see Ref. [

]. By inspecting the eﬀective mass plot of the propagator,

we ﬁx a suitable ﬁt range where the contributions of excited states is negligible and

4.3 Numerical study

0.05

0.1

0.15

0.2

0.25

−0.74 −0.72 −0.7 −0.68 −0.66 −0.64 −0.62

Pion mass m

Quark bare mass m

Pion mass m

at β = 6.14, 20

x 40 lattice

Wilson, linear fit

Wilson

Stag. Wilson, linear fit

Stag. Wilson

Figure 4.3.3:

Squared pion mass

as a function of the bare quark mass

with

a linear ﬁt through the data points with 0.05 ≤ m

≤ 0.15.

the signal-to-noise ratio is acceptable. We then do an uncorrelated ﬁt of the pion

propagator over the previously determined range using the function

(

)

= A cosh





−



(4.3.1)

by minimization of

, where

t = n

and the ﬁt quality varies in the range 0

χ/d.o.f. .

75. Here the ﬁt parameter

is the amplitude,

the pion mass and

∈

{

32, 40

}

is the extent of the corresponding lattice in temporal direction. The

errors are combined statistical and systematic errors and are determined with the

jackknife method [

–

], where we estimated the systematic error by varying the

range of the ﬁt.

In agreement with chiral perturbation theory, we ﬁnd an approximately linear

relationship between

and

. Moreover, the lowest achievable pion mass

for Wilson and staggered Wilson fermions is comparable. We also note that the

additive mass renormalization is smaller for staggered Wilson fermions compared

to Wilson fermions. This could be a (somewhat weak) indicator for a computational

advantage in the construction of overlap and domain wall fermions. Knowing

the relation between the pion mass and the bare quark mass, for a given value

we can use the respective bare mass

for Wilson and staggered Wilson

fermions. In the following, we ﬁx ﬁve values of the pion mass, namely

∈

{

0.04, 0.07, 0.10, 0.13, 0.16

}

, as in general the eﬃciency is a function of the ﬁxed

4 Computational eﬃciency

1.9 1.95 2 2.05 2.1 2.15 2.2 2.25

Number of configurations

(κ

/ κ

)

1/2

Distribution of condition number ratios

Figure 4.3.4:

Distribution of



/κ

for

10 on the 20

40 lattice at

β =

0.5

1.5

2.5

0.04 0.06 0.08 0.1 0.12 0.14 0.16

Wilson / stag. Wilson

Pion mass m

Cond. num. vs. CG iterations (averaged ratios, point sources)

1/2

ratio

CG ratio, ε = 10

−06

CG ratio, ε = 10

−10

CG ratio, ε = 10

−14

(a) Point sources

0.5

1.5

2.5

0.04 0.06 0.08 0.1 0.12 0.14 0.16

Wilson / stag. Wilson

Pion mass m

Cond. num. vs. CG iterations (averaged ratios, wall sources)

1/2

ratio

CG ratio, ε = 10

−06

CG ratio, ε = 10

−10

CG ratio, ε = 10

−14

(b) Wall sources

Figure 4.3.5:

Averages for the ratios of the number of iterations and



/κ

as a

function of the squared pion mass m

on a 16

× 32 lattice at β = 6.

physical parameter.

Using Chroma, we calculated the lowest and highest eigenvalue on each conﬁg-

urations to determine the conditions number

†

. With respect to Eq.

(4.2.6)

the ratio of the condition numbers is an important factor in judging the computa-

tional eﬃciency of staggered Wilson fermions. In Fig. 4.3.4, we give an example for

the distribution of



/κ

on one of our ensembles. As one can see, the square

root of the ratio is of order

(

)

and we conclude that the condition number of

Wilson fermions in typically a factor

(

)

higher compared to staggered Wilson

fermions.

In Figs. 4.3.5, 4.3.6 and 4.3.7, we ﬁnd the results for the ratio of the number of

4.3 Numerical study

0.5

1.5

2.5

0.04 0.06 0.08 0.1 0.12 0.14 0.16

Wilson / stag. Wilson

Pion mass m

Cond. num. vs. CG iterations (averaged ratios, point sources)

1/2

ratio

CG ratio, ε = 10

−06

CG ratio, ε = 10

−10

CG ratio, ε = 10

−14

(a) Point sources

0.5

1.5

2.5

0.04 0.06 0.08 0.1 0.12 0.14 0.16

Wilson / stag. Wilson

Pion mass m

Cond. num. vs. CG iterations (averaged ratios, wall sources)

1/2

ratio

CG ratio, ε = 10

−06

CG ratio, ε = 10

−10

CG ratio, ε = 10

−14

(b) Wall sources

Figure 4.3.6:

Averages for the ratios of the number of iterations and



/κ

as a

function of the squared pion mass m

on a 20

× 40 lattice at β = 6.

0.5

1.5

2.5

0.04 0.06 0.08 0.1 0.12 0.14 0.16

Wilson / stag. Wilson

Pion mass m

Cond. num. vs. CG iterations (averaged ratios, point sources)

1/2

ratio

CG ratio, ε = 10

−06

CG ratio, ε = 10

−10

CG ratio, ε = 10

−14

(a) Point sources

0.5

1.5

2.5

0.04 0.06 0.08 0.1 0.12 0.14 0.16

Wilson / stag. Wilson

Pion mass m

Cond. num. vs. CG iterations (averaged ratios, wall sources)

1/2

ratio

CG ratio, ε = 10

−06

CG ratio, ε = 10

−10

CG ratio, ε = 10

−14

(b) Wall sources

Figure 4.3.7:

Averages for the ratios of the number of iterations and



/κ

a function of the squared pion mass

on a 20

40 lattice at

β =

6.136716.

conjugate gradient iterations as given in Eq.

(4.2.6)

. In the ﬁgures, we show both

our theoretical estimate given by



/κ

and the average measured ratios for the

actual number of iterations needed as a function of the pion mass. We did this

measurements for diﬀerent values of the target residual

to check for a possible

dependence. We also used two diﬀerent kinds of sources, namely a point source

and a wall source.

First we note that the choice of the source has no noticeable eﬀect on the relative

computational performance as the ﬁgures are nearly indistinguishable. Remark-

ably the dependence of the ratios on

is also weak, although the number of

conjugate gradient iterations grows strongly with decreasing pion masses. We can

see that our rough theoretical estimate



/κ

is a decent approximation for

4 Computational eﬃciency

the actual measured ratios. Moreover, for smaller values of ε the measured ratios

become closer to our estimate as expected. In all cases we see that Wilson fermions

need roughly

(

)

more iterations in the application of the conjugate gradient

method, which is in agreement with Ref. [71].

If we increase the physical volume in our simulation, i.e. when we compare the

32 lattice at

β =

6 with the 20

40 lattice at

β =

6, we ﬁnd an improvement

at small

, but otherwise almost unchanged results. If, on the other hand, we

decrease the lattice spacing, i.e. when we compare the 16

32 lattice at

β =

with the 20

40 lattice at

β =

136716, we see a signiﬁcant improvement in

eﬃciency over the whole range of

. As the staggered Wilson Dirac operator is

more sensitive to gauge ﬂuctuations through its four-hop terms, we would expect

that on smoother gauge conﬁgurations its convergence properties improve more

compared to the Wilson Dirac operator.

4.3.2 Overall speedup

Combining the measured ratios from the previous discussion together with Eq.

(4.2.10)

the product in Eq. (4.2.4) takes the form

cost

(

#iterations

)

(

#iterations

)

(

cost mat-vec mult.

)

(

cost mat-vec mult.

)

≈ 2 × (2–3) = 4–6. (4.3.2)

This means that in our setting staggered Wilson fermions are a factor of 4

–

6 more

eﬃcient than usual Wilson fermions for inverting the Dirac operator on a source.

When computing the quark propagator, an additional factor of four should be

included in Eq.

(4.3.2)

as for Wilson fermions one needs four times as many sources.

Before we move on with the discussion of arithmetic intensities, we add some

critical remarks for the interpretation of the estimate given in Eq.

(4.3.2)

. Our cost

ratio is for the case of inverting the Dirac operator on a source and does not include

the setup phase. For staggered Wilson fermions, one would typically prepare the

averaged link products in the staggered Wilson term before applying the conjugate

gradient method. The factor in Eq.

(4.3.2)

would therefore be a bit lower when

taking the cost for this setup phase into account. Also, if one intends to do hadron

spectroscopy with staggered Wilson fermions as described in Chapter 5, one has

to take the additional number of sources needed into account, eﬀectively reducing

the computational advantage.

Moreover, when doing lattice ﬁeld theoretical computations on high-perfor-

4.4 Arithmetic intensities

mance computing clusters, communication over the network can become a bottle-

neck for the performance. As large-scale simulations of lattice quantum chromody-

namics usually make use of a domain-decomposition approach and the staggered

Wilson term connects each site with all the corners of the surrounding hypercube

via four-hop terms, it is expected that the resulting network traﬃc will be signiﬁ-

cant. How severely this limits the performance of staggered Wilson fermions in

large-scale simulations has yet to be seen. Until then, Eq.

(4.3.2)

should be taken

as an estimate for the achievable performance gain on shared memory machines.

4.4 Arithmetic intensities

On modern computer architectures the performance of lattice ﬁeld theoretical

simulations is often not limited by the ﬂoating point throughput, but by limited

memory bandwidth. A measure for the bandwidth requirements relative to the

number of ﬂoating point operations (FLOPs) is the so called arithmetic intensity

I =

FLOPs

Memory transactions in byte

. (4.4.1)

If for a given application this ratio is lower than what the respective hardware can

provide, characterized by

max. FLOPs / s

max. memory bandwidth in byte / s

, (4.4.2)

its performance is limited by the memory bandwidth. As a result, the computing

cores spend some time with idling while waiting for the completion of memory

access operations. On the other hand the performance of problems with high

arithmetic intensity is limited by the ﬂoating point throughput of the hardware.

Simulations in the context of lattice quantum chromodynamics tend to have

a rather low computational intensity, i.e. a relatively high number of memory

transactions per ﬂoating point operation. If optimized properly, lattice codes can

archive a high sustained performance on modern architectures. As the expected

performance depends on the arithmetic intensity of the lattice fermion formulation,

it is worth having a closer look.

In the following, we determine the arithmetic intensities for the application of

several lattice Dirac operators in the setting of quantum chromodynamics, where

we focus on the hopping-terms and exclude the trivial diagonal mass term. Within

4 Computational eﬃciency

derivations we abbreviate FLOPs with

. Before we begin, we note that for a com-

plex addition we need 2

and for a complex multiplication we need 6

. To multiply

a complex

(

)

matrix with a complex three-component vector, we need 66

. Re-

garding memory transactions, we assume single-precision ﬂoating point numbers,

so-called ﬂoats, which are 32

bit

or 4

byte

. Where convenient, we abbreviate ﬂoats

. Depending on the application and hardware, in practice one might also

use double-precision ﬂoating point numbers with 64

bit

or 8

byte

, lowering the

arithmetic intensity

by a factor of two (see Ref. [

]). In some cases even mixed-

precision is used, consisting of a combination of double-precision, single-precision

and half-precision ﬂoating point numbers.

4.4.1 Wilson fermions

We begin with Wilson fermions as deﬁned in Subsec. 2.2.6. We note that some

ﬁxed terms in the action can be precomputed, namely terms like

± γ

, to save

arithmetic operations. We also have to take into account that a competitive im-

plementation of Wilson fermions employs the spin-projection trick [

], which

reduces the number of spin components by a factor of two.

Floating point operations.

The spin projection takes 12

F =

(twelve

operations

four directions

forward/backward) and for the link multiplications

we need 66

F =

1056

(matrix-vector multiplication

two spin compo-

nents

four directions

forward/backward). Accumulating the resulting vector

takes 12

F =

168

(twelve components

cost per complex addition

adding

eight vectors). In total we ﬁnd that Wilson fermions take 1320

F/site

. This result is

in agreement with Refs. [92–94].

Memory access operations.

We need to read one spinor per direction, i.e. 12

fl =

192

(twelve complex numbers per spinor

ﬂoats per complex number

four directions

forward/backward) and one 3

3 gauge link per direction, i.e.

fl =

144

(nine elements

ﬂoats per complex number

four directions

forward/backward). Writing the resulting spinor takes 12

fl =

. In total

we ﬁnd 360

fl/site =

1440

byte/site

of memory access operations per lattice site.

This result is also in agreement with Refs. [92, 94].

While in this case the arithmetic intensity decreases, the number of CPU cycles can potentially

increase.

4.4 Arithmetic intensities

Arithmetic intensity. We ﬁnd

1320 FLOPs/site

1440 byte/site

≈ 0.92 FLOPs/byte (4.4.3)

for the arithmetic intensity of Wilson fermions.

4.4.2 Staggered fermions

We are repeating the previous exercise now for usual staggered fermions, which

we discussed in Subsec. 2.2.7.

Floating point operations.

For the link multiplications we have 66

F =

528

(matrix-vector multiplication

four directions

forward/backward). Accumulat-

ing the resulting vector takes 3

F =

(three color components

cost per

complex addition

adding eight vectors). In total we ﬁnd that usual staggered

fermions need 570 F/site, which is in agreement with Ref. [95].

Memory access operations.

We read one spinor per direction, i.e. 3

fl =

(three complex numbers per spinor

ﬂoats per complex number

four directions

forward/backward) as well as one gauge link per direction,

i.e. 9

fl =

144

(nine elements

ﬂoats per complex number

four

directions

forward/backward). Writing the resulting spinor takes 3

fl =

In total we have 198

fl/site =

792

byte/site

of memory access operations per lattice

site, which is also in agreement with Ref. [95].

Arithmetic intensity. We ﬁnd

570 FLOPs/site

792 byte/site

≈ 0.72 FLOPs/byte (4.4.4)

for the arithmetic intensity of staggered fermions.

4.4.3 Staggered Wilson fermions

Compared to usual staggered fermions, staggered Wilson fermions have an addi-

tional term in the action. In the following, we only analyze the additional contri-

butions from the staggered Wilson term and then add the respective counts to the

numbers of usual staggered fermions.

4 Computational eﬃciency

Floating point operations.

Applying the

operator takes 66

F +

F =

1146

(matrix-vector multiplication

number of hybercube corners

three color

components

cost per complex addition

adding sixteen vectors). Applying

and

takes

(

3 + 6

)

. The

term can be combined with the diagonal mass term

and, thus, will be ignored as in the preceding calculations. Accumulating the

resulting vector takes 3

F =

(three color components

cost per complex

addition

adding two vectors). In total we ﬁnd that staggered Wilson fermions

take an additional 1161

F/site

compared to staggered fermions, i.e. 1731

F/site

total. Note that when the diagonal

(

m + r

)

term is included we need additional

12 F, i.e. 1743 F/site in agreement with Ref. [4].

Memory access operations.

We need to read one additional spinor per hypercube

corner, i.e. 3

fl =

(three complex numbers per spinor

ﬂoats per

complex number

number of corners) and one gauge link per hypercube corner,

i.e. 9

fl =

288

(9 elements

ﬂoats per complex number

number of

corners) for the

operator. Write the resulting spinor takes 3

fl =

. This

means that we have an additional 390

fl/site =

1560

byte/site

, i.e. in total we ﬁnd

588 fl/site = 2352 byte/site for staggered Wilson fermions.

Arithmetic intensity. We ﬁnd

stw

1731 FLOPs/site

2352 byte/site

≈ 0.74 FLOPs/byte (4.4.5)

for the arithmetic intensity of staggered Wilson fermions.

4.4.4 Summary

We summarize our ﬁndings in Table 4.4.1, where we added Wilson fermions with

a clover term (see e.g. Ref. [

]) as well as HISQ [

] (highly improved staggered

quarks) and asqtad [

] (

-squared tadpole improved) staggered fermions from the

literature. We note that all staggered formulations have a very similar arithmetic

intensity, while staggered Wilson fermions have slightly lower memory bandwidth

requirements compared to the others. This means, that despite the large number

of memory access operations per site needed for the application of the staggered

Wilson Dirac operator, we do not expect a more severe memory bandwidth bottle-

neck.

4.5 Conclusions

Formulation Operations I/O Intensity Source

Wilson 1320 1440 0.92 Here and Refs. [92–94]

Wilson + Clover 1824 1728 1.06 Ref. [94]

Staggered 570 792 0.72 Here and Ref. [95]

HISQ / asqtad 1146 1560 0.73 Ref. [94]

Staggered Wilson 1731 2352 0.74 Here and Ref. [4]

Table 4.4.1:

Arithmetic intensities of several lattice Dirac operators. Operations are

in units of FLOPs / site, I/O in bytes / site and Intensity in FLOPs / byte.

One possibility to increase arithmetic intensities and to lower memory band-

width requirements for all fermion formulations is to only load the ﬁrst two rows of

each

(

)

matrix and then reconstruct the last one by using unitarity. This lowers

memory bandwidth requirements at the cost of additional FLOPs, thus increasing

arithmetic intensity.

4.5 Conclusions

We estimated the speedup factor of staggered Wilson fermions to be 4

–

6 compared

to usual Wilson fermions for inverting the Dirac matrix on a source. However, it is

important to keep the critical remarks from Subsec. 4.3.2 in mind when interpreting

this result.

To allow for an easier comparison, we discussed here the case of kernel operators

which are both unpreconditioned and unimproved. While for Wilson fermions

many preconditioned and improvement variants are known and in use, there are

many possibilities to further improve the computational eﬃciency of staggered

Wilson fermions as well. One could replace the usual staggered part of staggered

Wilson fermions by the HISQ action to reduce





eﬀects and use link smearing.

While a certain form of a clover term for

(

)

improvement was studied in Ref. [

there is also the possibility for the introduction of other clover terms.

Finally, comparing the arithmetic intensities of the diﬀerent fermion formula-

tions in Sec. 4.4, we note that the memory bandwidth requirements of staggered

Wilson fermions are slightly lower compared to other staggered formulations. As

memory bandwidth is usually a limiting factor for the performance of lattice codes,

4 Computational eﬃciency

this is important in the context of writing highly optimized implementations of

the Dirac operator.



C 

P  

One of the traditional applications of lattice gauge theory is hadron spectroscopy.

In this setting not only computational eﬃciency, but also the usability of the for-

malism is important. To show the feasibility of doing spectroscopy with staggered

Wilson fermions, in this chapter we discuss the case of pions, i.e. pseudoscalar

mesons.

For usual staggered fermions the four tastes give rise to sixteen pions. In the

following we study the pion spectrum of staggered fermions when turning Adams’

two-ﬂavor staggered Wilson term on. The eﬀect of this term is that two of the four

tastes acquire a mass of



−1



and become heavy. As a result, we ﬁnd that eight of

the pions become heavy as well, while the other eight remain light. In the physical

light part of the spectrum we ﬁnd the degenerate two-ﬂavor pion-triplet and the

ﬂavor-singlet

meson with each coming in two copies. The

meson remains light

as we omit the disconnected piece of the propagator in our study. Some of the

results discussed here were previously presented in Refs. [9, 10].

5.1 Introduction

From a practitioner’s point of view, the usability of a lattice formulation is important.

To show that staggered Wilson fermions are not at a disadvantage compared to

other fermion formulations, we illustrate spectroscopy calculations as one of the

traditional applications of lattice quantum chromodynamics (QCD). Like for other

staggered formulations, spin and ﬂavor degrees of freedom are distributed over

diﬀerent lattice sites. We will see, that one can easily adapt the spectroscopy

methods and operators of usual staggered fermions to the two ﬂavor case, as we

are going to illustrate for the case of pions.

In this work we deal with Adams’ original two ﬂavor proposal [

], where the

action reads

= ¯χ



+ m

+ W



χ, W

(

− Γ

)

(5.1.1)

5 Pseudoscalar meson spectrum

and

= η

∇

refers to the staggered Dirac operator,

to the bare quark mass,

to the staggered Wilson term as deﬁned in Sec. 3.4,

to the lattice spacing and

r >

0 is the equivalent of the Wilson parameter. Here and in the following, the

matrices refer to a representation of the Dirac algebra in ﬂavor space and in ﬁgures

we use the shorthand notation ξ

µν

≡ ξ

This chapter is organized as follows. In Sec. 5.2, we discuss pseudoscalar mesons

in the framework of usual staggered fermions and generalize the known methods

to the case of staggered Wilson fermions. In Sec. 5.3, we discuss our numerical

results and compare them to our theoretical predictions. Finally, in Sec. 5.4 we

summarize our results and end with some concluding remarks.

5.2 Pseudoscalar mesons

For usual staggered fermions, the meson and baryon rest-frame operators are

known [

] and we can adapt these operators to the two-ﬂavor staggered Wilson

case. However, as two of the four tastes become heavy after the introduction of

the staggered Wilson term, the physical interpretation of these operators change.

In the following, we will discuss the case of pseudoscalar mesons both from a

theoretical and numerical point of view. We ﬁrst begin with reviewing the case of

usual staggered fermions, before we move on to staggered Wilson fermions.

Staggered fermions have four (quark) tastes, giving rise to sixteen pseudoscalar

mesons. They have a spin ⊗ ﬂavor structure γ

⊗ ξ

with

∈



, ξ



. (5.2.1)

On the lattice the states fall into eight irreducible representations of the lattice

timeslice group [98], namely

, ξ

. (5.2.2)

Here the

⊗ ξ

state corresponds to the Goldstone pion for the axial symmetry at

vanishing

. While for staggered fermions the continuum

(

)

ﬂavor symmetry

is broken down to a discrete subgroup, by doing a joint expansion in

and

one

can show that the pion masses are

(

)

symmetric up to leading discretization

errors, see Ref. [99].

Within these representations we ﬁnd two distinct types of states: while some

states propagate with a

(

−1

)

-factor in time, others do not. The timeslice operators,

5.2 Pseudoscalar mesons

Flavor matrix ξ

Composition Particles

qσ

q π triplet

qq η singlet

Table 5.2.1:

Pseudoscalar meson spectrum for staggered Wilson fermions (

and

k mutually distinct).

in general, excite both these kinds of states, i.e. if an operator couples to

⊗ ξ

then it also couples to

⊗ξ

. In the case of pseudoscalar mesons we have

∈

{

, γ

}

. The fact that we excite two states simultaneously does not pose a

problem as both states can be easily separated by their distinct time-dependence.

We can parametrize the time-time correlation function by

(

)

= R

cosh





−



(

−1

)

−

cosh



−



−



, (5.2.3)

where

t = n

and we assume that one-particle states are dominant [

]. Here

and

are the amplitudes and masses of both respective states and

refers to

the number of timeslices of the lattice.

After the introduction of the staggered Wilson term, two tastes become heavy

and ξ

can be thought of having the structure





light + light light + heavy

heavy + light heavy + heavy



. (5.2.4)

In the continuum limit, the heavy quark content decouples and the “light + light”

part of ξ

determines the physical interpretation. To give an example, consider

= ξ



iσ



. (5.2.5)

The physically relevant part is

iσ

, hence

−i ¯χ

(

⊗ ξ

)

corresponds to a

operator, see also Ref. [9].

After analyzing all states, one ﬁnds that for

∈

{

, ξ

}

there are

only contributions from heavy tastes. For the states with

∈



, ξ



we ﬁnd a mixture of purely light and purely heavy quark content, where the heavy

5 Pseudoscalar meson spectrum

0.05

0.1

0.15

0.2

0.25

0 0.005 0.01 0.015 0.02 0.025 0.03

Quark bare mass m

Our γ

x ξ

Our γ

x ξ

Our γ

x ξ

Our γ

x ξ

Lit. γ

x ξ

Lit. γ

x ξ

Lit. γ

x ξ

Lit. γ

x ξ

(a) Part I

0.05

0.1

0.15

0.2

0.25

0 0.005 0.01 0.015 0.02 0.025 0.03

Quark bare mass m

Our γ

x 1

Our γ

x ξ

Our γ

x ξ

Our γ

x ξ

Lit. γ

x 1

Lit. γ

x ξ

Lit. γ

x ξ

Lit. γ

x ξ

(b) Part II

Figure 5.3.1:

Comparison between the pseudoscalar meson masses





r =

0 in our study with the literature values from Ref. [100].

0.1

0.2

0.3

0.4

0.5

0.6

−0.065 −0.06 −0.055 −0.05 −0.045 −0.04 −0.035 −0.03 −0.025 −0.02

Quark bare mass m

x ξ

x 1

x ξ

(a) Physical pseudoscalar mesons

0.1

0.2

0.3

0.4

0.5

0.6

−0.065 −0.06 −0.055 −0.05 −0.045 −0.04 −0.035 −0.03 −0.025 −0.02

Quark bare mass m

x ξ

(b) Heavy pseudoscalar mesons

Figure 5.3.2: Pseudoscalar meson masses m





at r = 0. 1.

contributions decouple in the continuum limit. In total we ﬁnd that from the

sixteen pseudoscalar mesons for usual staggered fermions only eight remain light

in the presence of the staggered Wilson term. Among these eight light pions we

ﬁnd two copies of the two-ﬂavor pion triplet, which only diﬀer by their heavy quark

content. We note that the pion triplet for staggered Wilson fermions is degenerated

as discussed in Ref. [

]. In addition we ﬁnd two copies of the ﬂavor-singlet

meson. These light states and their physical interpretation are summarized in

Table 5.2.1, where q =

(

, q

)

denotes the two physical tastes.

5.3 Numerical results

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

10 20 30 40 50

Time slice t

Staggered Wilson fermions, ξ

= ξ

, β = 6, 16

x 64, m

= −0.6650

Fit: m = 0.511138 ± 0.001407 ± 0.000469, χ

/d.o.f. = 0.4704

Figure 5.3.3:

Eﬀective mass plot

eff

(

)

together with the ﬁtted mass

and its

statistical and systematic error at

r =

1 for the light pseudoscalar

meson with ξ

= ξ

5.3 Numerical results

To illustrate the practicality of spectrum calculations with staggered Wilson fermi-

ons and to verify the results from the previous section, we implemented staggered

Wilson fermions in the Chroma/QDP software package [

]. The software suite

comes with an application for spectrum calculations with staggered fermions

For the pseudoscalar mesons operators we used the operators that are already

implemented and which match the ones used in Ref. [100].

Our quenched QCD study was carried out using 200 gauge conﬁgurations at

a coupling of

β =

6 on a 16

64 lattice, where we ﬁxed the conﬁgurations to

Coulomb gauge. On each conﬁguration we evaluated eight staggered wall sources

i.e. wall sources where within the wall there is a one in a single corner of each of the

spatial cubes and zero everywhere else. By combining them appropriately, we can

project onto the irreducible representations of the timeslice group [

100

]. To extract

the pseudoscalar meson masses, we essentially follow the method described in

Subsec. 4.3.1. To compute eﬀective masses, we can do a minimization of the form

The program spectrum_s.

Note that this additional number of sources eﬀectively reduces the computational speedup factor

as presented in Chapter 4.

5 Pseudoscalar meson spectrum

0.05

0.1

0.15

0.2

0.25

0.3

−0.74 −0.73 −0.72 −0.71 −0.7 −0.69 −0.68 −0.67 −0.66

Pseudoscalar meson mass m

Quark bare mass m

x ξ

x 1

x ξ

Figure 5.3.4:

Pseudoscalar meson spectrum for two-ﬂavor staggered Wilson fermi-

ons (r = 1).

min



with







(

+ 0

)

− p

(

+ 0

)

(

+ 1

)

− p

(

+ 1

)

(

+ 2

)

− p

(

+ 2

)

(

+ 3

)

− p

(

+ 3

)







, (5.3.1)

where

(

)

refers to the ﬁt function deﬁned in Eq.

(5.2.3)

and

(

)

to the

corresponding time-time correlation function. In the ﬁnal four parameter ﬁt of

Eq.

(5.2.3)

we ﬁxed the ﬁt range from

10 to

25 together with the mirrored

range

− n

for all pions, where

64 is the extent of the lattice

in temporal direction. Correlations are neglected in the

-ﬁt. The errors are

estimated in the same way as described in Subsec. 4.3.1.

As a check of the correctness of our method and ﬁtting procedure, we reproduced

the known results from Ref. [

100

] (cf. Ref. [

101

]) for the case of usual staggered

fermions. By setting the staggered Wilson parameter to

r =

0 in Eq.

(5.1.1)

for this

test, we can calculate the pseudoscalar meson spectrum and ﬁnd an excellent

match with the literature values, see Fig. 5.3.1. In this ﬁgure and the following

mass plots, for every pseudoscalar meson mass

we add a linear ﬁt of the form

= c

+ c

As the negative ﬂavor-chirality species acquire a mass of

(

r /a

)

in the presence

5.4 Conclusions

of the staggered Wilson term, we can use

to control the masses of the heavy

pseudoscalar mesons. We can then consider staggered Wilson fermions with a

small

to observe how a mass splitting between the light and heavy pions forms.

In Fig. 5.3.2, we can ﬁnd the spectrum for

r =

1, where in the left part of the ﬁgure

we can ﬁnd all light pions and in the right part the heavy pions. Using the same

scale for the vertical axes, one can see how the heavy pseudoscalar mesons are

systematically shifted upwards due to contributions from heavy tastes.

Moving on to staggered Wilson fermions with

r =

1, the initially small mass gap

between the light and heavy pion states becomes large. In Fig. 5.3.3, we can see a

representative eﬀective mass plot for the state with

= ξ

, which remains light

after the introduction of the staggered Wilson term in agreement with Table 5.2.1

and the discussion in Sec. 5.2. On the other hand, the heavy states now have a

mass of

m = O



−1



. As the condition

am 

1 is violated, one would not expect

to be able to reliably describe these states on our lattices.

Finally, focusing on the physical (light) part of the pseudoscalar mesons, we ﬁnd

the resulting spectrum of pseudoscalar mesons for two-ﬂavor staggered Wilson

fermions in Fig. 5.3.4. As predicted by chiral perturbation theory, we ﬁnd that the

squared pseudoscalar meson masses

are approximately linear in the bare quark

mass

. Referring to Table 5.2.1, in the continuum limit the pairs

{

}

and



, ξ



are expected to be degenerated as the heavy quark content decou-

ples. In the present setting with a ﬁnite lattice spacing these degeneracies only

hold approximately, but already describe the observed spectrum well. Moreover,

we observe that both pairs separate well. We note that in our calculations the

ﬂavor-singlet

remains light due to the omission of the disconnected piece of the

propagator.

5.4 Conclusions

The aim of this study was to demonstrate the feasibility of spectrum calculations

with Adams’ two-ﬂavor staggered Wilson fermions. To this end we determined how

the pseudoscalar spectrum can be computed and changes with the presence of

the staggered Wilson term.

In the physical part of the spectrum we ﬁnd two copies for both the degenerate

two-ﬂavor pion-triplet as well as the ﬂavor-singlet

meson. We conclude that

two-ﬂavor staggered Wilson fermions work as expected for studying pseudoscalar

5 Pseudoscalar meson spectrum

mesons. In future work one can investigate spectroscopy of other hadrons, in

particular also baryons, where our approach is expected to generalize.



C 

S  

Chiral symmetry plays an important role in understanding the low-energy regime

of quantum chromodynamics, in particular hadron phenomenology. As discussed

in Subsec. 2.2.4, the Nielsen-Ninomiya theorem [

–

] complicates the implemen-

tation of chiral symmetry on the lattice. With the overlap construction [

102

–

108

] this

problem could eventually be overcome and one can formulate a lattice Dirac oper-

ator with exact chiral symmetry [

109

–

111

]. While the overlap formalism has many

attractive theoretical properties, its practical applications are limited due to the fact

that overlap fermions typically require a factor of O

(

10–100

)

more computational

resources compared to Wilson fermions.

The prospect of a staggered overlap operator becomes interesting as it can po-

tentially bring down the enormous costs of lattice theoretical simulations in which

chiral symmetry is of importance. Usual staggered fermions do not deﬁne a suit-

able kernel operator for the overlap formalism due to the fact that the staggered

equivalent of

, namely the

 γ

⊗

+ O





operator, does not square to

the identity, i.e.

. As Adams pointed out in Refs. [

112

], this problem

can be solved with the help of the staggered Wilson construction, where on the

physical species

can be used for

. As

holds exactly, this allows the

construction of staggered variants of overlap and domain wall fermions.

In this chapter, we discuss Neuberger’s overlap fermions in Sec. 6.1 and then

introduce Adams’ staggered overlap construction in Sec. 6.2. This is followed by a

review of the index theorem in Sec. 6.3. In Sec. 6.4, we verify that the continuum

limit of the index and axial anomaly for the staggered overlap Dirac operator is cor-

rectly reproduced. This is meant as a ﬁrst step to place staggered overlap fermions

on solid theoretical foundations. Our analytical work was ﬁrst presented in Ref. [

where later a more comprehensive discussion was given by Har in Ref. [11].

6 Staggered overlap fermions

6.1 Neuberger’s overlap fermions

We begin by discussing the Ginsparg-Wilson relation in Subsec. 6.1.1 and Neu-

berger’s overlap construction in Subsec. 6.1.2, following the outline of Ref. [16].

6.1.1 The Ginsparg-Wilson relation

The Nielsen-Ninomiya theorem prevents a straight-forward implementation of

chiral symmetry on the lattice of the form

{

D, γ

}

0. To circumvent this problem,

Ginsparg and Wilson proposed in Ref. [

109

] to replace the continuum expression

{

D, γ

}

= aD γ

D. (6.1.1)

For this so-called Ginsparg-Wilson relation in Eq.

(6.1.1)

we see that the anticommu-

tator is not vanishing on the lattice, but is instead an order

(

)

term, eventually

shrinking to zero in the continuum limit. A remarkable observation is now that,

while the continuum form of chiral symmetry is recovered in the

a →

0 limit,

even at a ﬁnite lattice spacing a Dirac operator

satisfying Eq.

(6.1.1)

has an exact

chiral symmetry. However, for a long time no solution to the Ginsparg-Wilson rela-

tion was known and the possibility of ﬁnding a closed solution in the interacting

case was doubted. Only much later solutions were explicitly constructed, namely

overlap fermions [108] and ﬁxed points fermions [113].

Assuming that a Dirac operator

satisﬁes the Ginsparg-Wilson relation, one

can deﬁne an inﬁnitesimal chiral transformation of the form

δΨ = Ψγ

, δΨ = γ

(

− aD

)

Ψ, (6.1.2)

where the corresponding ﬁnite transformation takes the form

Ψ → Ψe

iαγ

, Ψ → e

iαγ

(

−aD

)

Ψ (6.1.3)

with

α ∈ R

being a continuous parameter. We note that these transformations can

also be written in a symmetric form, where both

and

transform with a factor

(

− aD/2

)

. One can easily verify the invariance of the action

S = ΨD Ψ

under

these chiral transformations, where in the

a →

0 limit the continuum expressions

are reproduced.

Now let us impose that

is also

Hermitian, i.e.

†

= γ

Dγ

. This ensures that

complex eigenvalues come in conjugated pairs and

det D ∈ R

. Another important

6.1 Neuberger’s overlap fermions

consequence is that for an eigenvector

, corresponding to the eigenvalue

one can verify that

, γ

= 0, (6.1.4)

unless

λ ∈ R

. Here

v, w

= v

†

denotes the usual inner product of

. This

means that only eigenmodes with real eigenvalues can have deﬁnite chirality.

Going back to the Ginsparg-Wilson relation in Eq.

(6.1.1)

, we can see that by

multiplying with

from either the left or the right side and using

Hermiticity,

one ﬁnds that

†

D = D

†

+ D = D + D

†

= aDD

†

, (6.1.5)

so D is a normal operator satisfying



†

, D



= 0. As a consequence one ﬁnds

λ + λ



= a λλ



(6.1.6)

for the eigenvalues. Writing the eigenvalues explicitly with real and imaginary part,

this equation takes the form



Re λ −



(

Im λ

)

(6.1.7)

and one can see that they lie on a circle in the complex plane, commonly referred

to as the Ginsparg-Wilson circle.

As the Ginsparg-Wilson circle is going through the origin,

can have exact zero

modes. If v

denotes such a zero mode, one can verify that

= γ

= D γ

= 0. (6.1.8)

Thus on the eigenspace for the eigenvalue zero the Dirac operator

and the

chirality operator

commute. One can now choose the eigenvectors so that they

are simultaneously eigenmodes of both D and γ

. We can then write

= ±v

, (6.1.9)

where the eigenvalues are

1 due to

. This means that the zero modes of

have deﬁnite chirality. If we ﬁnd a plus sign in Eq.

(6.1.9)

, we say that

has positive

chirality or call it right-handed, otherwise we say that

has negative chirality or

call it left-handed.

6 Staggered overlap fermions

6.1.2 The overlap operator

While for a long time the search for an explicit solution to the Ginsparg-Wilson

relation was not successful, eventually a solution in the form of the overlap operator

was found. The overlap formalism was developed over the course of several papers

[

102

–

108

] and ﬁnally brought into its present day form by Neuberger [

106

–

108

]. The

Dirac operator reads



+ A



†



−





+ γ

H /

√



(

+ γ

sign H

)

(6.1.10)

with kernel operator

A = D

(

−M

)

, Hermitian operator

H = γ

(

−M

)

and

negative mass parameter

. In the free-ﬁeld case valid choices are in the range

∈

(

0, 2r

)

, where

denotes the Wilson parameter, to give rise to a one ﬂavor

theory. In some contexts there is an additional rescaling of

by some factor

see our discussion in Chapter 8. In principle one can replace the Wilson Dirac

operator

also by some other suitable

Hermitian kernel operator. Using the

fact that

sign

H =

, one can easily verify that the overlap Dirac operator satisﬁes

the Ginsparg-Wilson relation in Eq. (6.1.1).

The overlap Dirac operator in Eq.

(6.1.10)

is not ultralocal due to the presence

of the 1

√

term. Here we call a lattice Dirac operator

ultralocal if

∃l ≥

0 so

that

(

x, α, a; y, β, b

0 for

x − y

> l

, where

(

x, α, a; y, β, b

)

denotes

the matrix element of

connecting the space-time coordinates

and

with

respective spin components

and color components

. Under some general

assumptions this lack of ultralocality holds in fact for all solutions of the Ginsparg-

Wilson relation [

114

]. However, locality of

could be established in Ref. [

115

] in

the form of

(

x, α, a; y, β, b

≤ λ e

−γ/a·

x−y

. (6.1.11)

Here

λ ≥

0 and

γ >

0 are constants independent of the gauge ﬁeld. As a result, the

decay constant

a/γ

vanishes in the continuum limit

a →

0 and a local quantum

ﬁeld theory is recovered.

The presence of the

sign

-function in Eq.

(6.1.10)

makes the application of the

overlap Dirac operator computational expensive. As mentioned earlier, one typ-

ically needs an factor of

(

10–100

)

more computational resources compared to

Wilson fermions. As a consequence the use of overlap fermions is limited in practi-

cal applications and a lot of past work deals with the eﬃcient numerical treatment

of the sign-function, see e.g. Refs. [116–120].

6.2 Staggered overlap fermions

When applying the overlap construction to the staggered Dirac operator, one en-

counters some severe problems. For the resulting operator the remnant chiral

symmetry is lost, the Ginsparg-Wilson relation does not hold and exact zero modes

are absent [45].

These problems can be traced back to the fact that the overlap construction

relies on the property

, while this does not hold for the staggered equivalent

 γ

⊗

+ O





. The eigenvalues of

are

1, while the eigenvalues of

are

distributed throughout the interval

[

−1, 1

]

and include zero as can be seen from

the momentum representation



cos

in the free-ﬁeld case [

112

]. Finally,

we note that the operator

, unlike its Wilson equivalent

, is even lacking

Hermiticity [

112

]. For these reasons the overlap construction with a usual staggered

kernel does not give rise to a proper lattice fermion formulation.

All these problems could be eventually overcome by Adams’ construction in-

troduced in Refs. [

112

]. The idea is that, after the introduction of the stag-

gered Wilson term, one can use the

operator instead of

. Although

has

a spin

⊗

ﬂavor interpretation [

] of

 γ

⊗ ξ

, the staggered Wilson term

projects out the negative ﬂavor-chirality species and we ﬁnd

 Γ

+ O





the physical species. Furthermore,

holds exactly and

= Γ

deﬁnes

a Hermitian operator.

As a consequence, one can apply the overlap construction to the staggered

Wilson kernel with the help of a simple replacement rule given by

→ D

, γ

→ Γ

(6.2.1)

as derived in Refs. [

112

]. Explicitly, the staggered overlap operator

reads



+ A



†



−





+ Γ





(

+ Γ

sign H

)

(6.2.2)

with kernel operator

A = D

(

−M

)

, Hermitian operator

H = Γ

(

−M

)

and

negative mass parameter

within the range

∈

(

0, 2r

)

in the free-ﬁeld case. We

ﬁnd that the action

= χD

is invariant under the inﬁnitesimal transformation

δ χ = χ Γ

, δ χ = Γ

(

− aD

)

χ, (6.2.3)

cf. Eq.

(6.1.2)

for the usual overlap operator. Furthermore, the staggered overlap

6 Staggered overlap fermions

operator satisﬁes the Ginsparg-Wilson relation

{

, D

}

. (6.2.4)

As in four space-time dimensions

describes two tastes, the staggered overlap

operator

represents two degenerate chiral fermion species on the lattice, which

can be used to describe e.g. the up and down quark.

6.3 The index theorem and the axial anomaly

With respect to the topological aspects we are discussing throughout this thesis,

let us introduce some important concepts from topology. If

∓

denote the number

of left-handed and right-handed zero modes, then we deﬁne the integer-valued

index of the Dirac operator D as

index D = n

−

− n

. (6.3.1)

In the

-dimensional (

even) continuum setting of quantum chromodynamics,

the famous Atiyah-Singer index theorem [121–124] states that

−

− n

(

−1

)

d/2

cont

, (6.3.2)

where Q

cont

is the so-called topological charge given by

cont



x q

cont

(

)

, q

cont

(

)

32π

µν ρσ



µν

(

)

ρσ

(

)



(6.3.3)

and

µν

(

)

is the corresponding ﬁeld strength tensor. The index theorem in Eq.

(6.3.2)

is remarkable given the fact that the right hand side appears to be a continuous

function of the gauge ﬁeld, while the left hand side is an integer number. On the

lattice one can formulate a similar index theorem [110], which reads

−

− n

(

−1

)

d/2

Q (6.3.4)

with

Q = a



x ∈Λ

(

)

, q

(

)

[

(

x, x

)]

. (6.3.5)

We note that in practice one never ﬁnds both

−

and

to be non-vanishing at

the same time on thermalized conﬁgurations, i.e. one ﬁnds either

−

(

−1

)

d/2

(

−1

)

1+d/2

. Moreover, in two dimensions one can show the Vanishing

6.4 Continuum limit of the index and axial anomaly

Theorem [

125

–

127

]. That is, one can prove that if

Q ,

0, then either

−

necessarily vanishes.

While a chiral ﬂavor-singlet rotation is a symmetry of the action for a Dirac

operator satisfying the Ginsparg-Wilson relation, the path integral itself is not

invariant due to the nontrivial transformation of the integration measure [

110

111

cf. Refs. [

128

–

132

] for the continuum case. This so-called axial anomaly results in an

anomalous continuity equation for the singlet axial vector current. The presence

of the axial anomaly implies that there is no ﬂavor-singlet Goldstone particle and

explains why the

-meson is signiﬁcantly heavier than the pions assuming an

approximate chiral symmetry for two ﬂavors, see e.g. Ref. [

]. In the three ﬂavor

case the Witten-Veneziano formula [

133

–

140

] connects the mass of the

-meson to

the topological susceptibility χ

top





/V , where V is the space-time volume.

6.4 Continuum limit of the index and axial anomaly

Massless overlap fermions have exact zero modes with deﬁnite chirality and one

can deﬁne an index for the overlap Dirac operator as discussed in Subsec. 6.3.

Overlap fermions implement an exact chiral symmetry on the lattice, which is

anomalously broken in the quantum case and gives rise to the axial anomaly. This

axial anomaly and the index density of the overlap Dirac operator are proportional,

while the continuum axial anomaly is proportional to the topological charge density

and by the Atiyah-Singer index theorem [

121

–

124

] we ﬁnd equality between the

topological charge and the index.

In the setting of lattice quantum chromodynamics a natural question is then if

the continuum axial anomaly and index are recovered in the continuum limit in

suﬃciently smooth gauge background ﬁelds. In the past several investigations [

109

110

141

–

145

] dealt with this question. Here, as a ﬁrst step to secure the theoretical

foundations for Adams’ construction, we want to check that the continuum axial

anomaly and index is correctly reproduced for the staggered overlap Dirac operator.

Our strategy for evaluating the continuum limit is similar to the Wilson case [

144

145

], although one encounters some technical complications due to the nontrivial

spin-ﬂavor structure of staggered fermions. In Ref. [

], we found that the index

correctly reproduces the index in the continuum (cf. the numerical studies in

Refs. [

146

]), while for the anomaly ﬁrst an averaging over the sites of a lattice

hypercube is needed. This is not unexpected as in the staggered formalism spin

6 Staggered overlap fermions

and ﬂavor degrees of freedom are distributed over lattice hypercubes, while the

distance between neighboring lattice sites is of order

(

)

, shrinking to zero in

the continuum limit.

In this section, we brieﬂy review our discussion

given in Ref. [

], where we

previously presented the following results. For a more elaborate discussion, we

refer the reader to the thesis in Ref. [11].

6.4.1 The index and axial anomaly

We now give an overview of the derivation of the continuum limit of the index

and anomaly of the staggered overlap operator in the setting of lattice quantum

chromodynamics. We begin by recalling the deﬁnition of the staggered overlap

operator as deﬁned in Eq. (6.2.2) for the case of r = 1, which reads



+ Γ

√



, H = Γ

(

−m

)

. (6.4.1)

Using the index formula in Ref. [110], we can write

index D

= −



√



. (6.4.2)

In the following, our setting is a four-dimensional box of length

with lattice

spacing

and

sites in each direction. The lattice transcripts of a smooth con-

tinuum gauge ﬁeld

(

)

are taken as the link variables, taking values in the Lie

algebra

(

)

(

)

. Furthermore, we impose boundary conditions as described

in Ref. [145].

Let us now express Eq. (6.4.2) as a sum over the index density

index D

= a



(

)

, q

(

)

= −



√



(

x, x

)

. (6.4.3)

Here

denotes a color trace and

(

x, y

)

is deﬁned for an operator

Θ χ

(

)



(

x, y

)

(

)

. Like in the case of usual overlap fermions, we ﬁnd that the index

density

(

)

is related to the axial anomaly via

(

)

(

)

, where

(

)

is the

divergence of the axial current.

Discussion based on D. H. Adams, R. Har, Y. Jia, and C. Zielinski, “Continuum limit of the axial

anomaly and index for the staggered overlap Dirac operator: An overview,” PoS

LATTICE2013

(2014) 462,

arXiv:1312.7230 [hep-lat]

. Copyright owned by the authors under the terms of

the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License

(CC BY-NC-ND 4.0).

6.4 Continuum limit of the index and axial anomaly

In the case of the overlap construction with a Wilson kernel, the central problem

in verifying the correct continuum limits of the index and axial anomaly is to show

that

lim

a→0

(

)

= −

32π

µνσ ρ



µν

(

)

σ ρ

(

)



, (6.4.4)

where

denotes a trace in color space, as it was done in Refs. [

144

145

]. The

a →

limit is taken by consecutive symmetric reﬁnements of the underlying lattice. As

mentioned earlier, in the case of the staggered overlap operator Eq.

(6.4.4)

only

holds if we average over the sites of a lattice hypercube which contains

. For the

remainder of this section we are going to outline the proof of Eq.

(6.4.4)

, where we

have to include the averaging procedure and a factor of two on the right hand side

to account for the two physical tastes of the staggered Wilson kernel.

6.4.2 Continuum limit of the index density

Following the derivation for the usual overlap case given in Ref. [

145

], we use the

identity

lim

a→0

√

(

x, x

)

= lim

a→0

3π/2a



−π/2a

p e

−ipx

√

ipx

, (6.4.5)

where expressions like

are understood as scalar products. Let us begin by intro-

ducing some useful notation. We write the lattice momentum

∈

[

−π/2, 3π/2

]

+ q

, where

∈

{

0, 1

}

and

∈

[

−π/2, π/2

]

. Furthermore, we deﬁne

the sum of two vectors

A, B ∈

{

0, 1

}

as the componentwise sum

mod

2 and let

(

)

≡ exp

(

iπBx/a

)

for

B ∈

{

0, 1

}

. On the vector space

spanned by the

, we

can deﬁne two commuting representations of the Dirac algebra [42] given by





(

−1

)

A,B+η

, (6.4.6)





(

−1

)

A,B+ζ

, (6.4.7)

where





1 for

σ < µ





1 for

σ > µ

and all remaining components

being zero. Here we use the notation with hat as the versions without the hat

denote the extensions from the space

to the one-component spinor ﬁeld space.

In the following, we use the notation

≡ Γ

, C

≡



sym

α βγδ

/4!, (6.4.8)

where

is the symmetric product of the



+ T

†



2 operators. In our

6 Staggered overlap fermions

notation, the staggered Wilson term reads

(

− Ξ

)

, cf. Sec. 3.4. Writing a

plane wave as

exp

(

ipx

)

= exp

(

[

πB/a + q

]

)

= e

(

)

exp

(

iqx

)

, (6.4.9)

we can understand the action of the staggered Wilson kernel D

in AB-language

to be

ipx

= D



(

)

iqx



= e

(

)







∇



−







iqx

= e

(

)



∇



−





iqx

≡ e

(

)





iqx

, (6.4.10)

where we sum over all sixteen possible values of the repeated

-index. Similarly,

we ﬁnd for the Γ

operator

ipx

= Γ



(

)

iqx



= e

(

)





iqx

. (6.4.11)

We then conclude that

√

ipx

√



(

)

iqx



= e

(

)



√



iqx

, (6.4.12)

where

follows from

by replacing

→

and

→

. This allows us

to rewrite Eq. (6.4.5) as

lim

a→0

√

(

x, x

)

= lim

a→0



π/2a



−π/2a

q e

−iqx

(

)

√

(

)

iqx

= lim

a→0



A,B

iπ

(

A−B

)

x/a

π/2a



−π/2a

q e

−iqx



√



iqx

.(6.4.13)

In the last sum in Eq.

(6.4.13)

, we now separately analyze the contributions from

the case

A = B

and

A , B

. While the former reproduces the continuum topological

charge density

(

)

, the latter contributions vanish after averaging over a lattice

hypercube containing x .

6.4 Continuum limit of the index and axial anomaly

The A = B case

Let us begin with the A = B case in Eq. (6.4.13). In this case we ﬁnd

lim

a→0

π/2a



−π/2a

q e

−iqx



√



iqx

, (6.4.14)

where the trace is taken in the vector space

. Following Ref. [

], there exist an

isomorphism such that

 γ

⊗

and



⊗ξ

. In a basis where

is diagonal,

we ﬁnd



⊗



∇

(

⊗

)

(

1 ∓C

)

(6.4.15)

with signs

∓

for the ﬂavors on which

= ±

. On each of these two ﬂavor subspaces

is a hypercubic lattice Dirac operator as discussed in Ref. [

147

] and



⊗ ξ

= ±

(

⊗ 1

)

. Noting that the momentum representation of

in the free-

ﬁeld case reads

(

)



cos





= 1 + O





, (6.4.16)

we see that

describes a single physical ﬂavor for each of the two species of

the subspace where

. Finally, note that in this case Eq.

(6.4.14)

equals



H /

√



(

x, x

)

with

H =

(

⊗

)



− m



. Using the general result of

Ref. [

147

] and the fact that the trace over the ﬂavor subspace produces a factor of

two, we ﬁnd that

(

)

yields the continuum topological charge density with an

additional factor of two due to the two physical species of the staggered Wilson

kernel.

At the same time, the contribution from the other ﬂavor subspace, where

−

, vanishes as there are no physical fermion species, which again follows from

Ref. [147].

The A , B case

We are left with analyzing the

A , B

contributions in Eq.

(6.4.13)

when using the

mentioned averaging procedure. Rewriting the prefactor

iπ

(

A−B

)

x/a

(

−1

)

(

A−B

)

with

n = x /a

, we see that the sum over lattice sites gives zero if not all components

of A −B are zero. Hence the problem is reduced to showing that

π/2a



−π/2a

q e

−iqx



√



iqx

(6.4.17)

6 Staggered overlap fermions

changes by order

(

)

when

is moved within a lattice hypercube. This can be

shown by expanding the integrand in powers of the continuum gauge ﬁeld as done

in Refs. [

144

145

]. While the integral formally diverges as



−4



, one can show

that the lowest non-vanishing contribution in the expansion is of order





. As

the smooth continuum gauge ﬁeld varies by

(

)

when moving

inside a lattice

hypercube, we conclude that Eq.

(6.4.17)

changes by an order

(

)

term as well,

completing our proof.

Further discussion

For more details on this derivation we refer the reader to Refs. [

]. While Ref. [

]

deals with the above derivation, the thesis in Ref. [

] also discusses the index of

lattice Dirac operators in a wider setting.



C 

S 

An important motivation for studying staggered overlap fermions [

112

] is

the prospect of bringing down the enormous computational costs of simulations

with lattice fermions with exact chiral symmetry. A ﬁrst study on this was done in

Refs. [

], where in the free-ﬁeld limit a large speedup factor of

(

)

compared

to usual overlap fermions was found. However, after the introduction of a ther-

malized QCD background ﬁeld this factor dropped to

(

)

. At the same time, we

observed a large speedup factor of 4

–

6 for the staggered Wilson kernel compared

to the Wilson kernel as discussed in Chapter 4.

In this chapter, we try to explain the somewhat surprising observation that

the overlap construction shows a much smaller speedup factor compared to the

underlying kernel operator. We try to explain this discrepancy with the spectral and

chiral properties of the staggered Wilson Dirac operator for diﬀerent lattices sizes.

To this end, we note that the studies of staggered overlap fermions in Refs. [

]

were carried out on relatively small lattices, where the eigenvalue spectrum has a

small gap and diﬀuse branches. Our investigations of the computational eﬃciency

of the staggered Wilson kernel operator in Chapter 4, on the other hand, were done

on signiﬁcantly larger lattices.

In the following, we present indicators in favor of the idea that the spectral and

chiral properties of the staggered Wilson kernel improve on larger lattices and, thus,

the performance of staggered overlap fermions on small lattices is not representa-

tive of the expected performance in a realistic setting. As only an actual benchmark

on larger lattices will eventually clarify this point, the following discussion should

be taken as a proposed explanation. Some of the results discussed in this chapter

were previously presented in Refs. [4, 7].

7.1 Introduction

While usual overlap fermions [

102

–

108

] are very attractive due to the presence of

chiral zero modes and having a well-deﬁned index, their high computational costs

7 Spectral properties

make their practical use for high-precision studies limited. By using staggered

overlap fermions, which use the staggered Wilson Dirac operator as a kernel oper-

ator, one can potentially reduce these costs signiﬁcantly. Initial numerical tests

conﬁrmed this expectation only partially [

], see also Ref. [

]. While in the

free-ﬁeld case an impressive speedup factor of

(

)

compared to usual overlap

fermions was observed, in a quenched QCD background ﬁeld this factor reduced

to O

(

)

Naturally the computational performance of the overlap operator depends on

the spectral and chiral properties of the underlying kernel operator. In particular,

a more chiral kernel is expected to result in lowered computational costs [

148

]. The

reduction of the speedup factor for staggered overlap fermions, when moving from

the free-ﬁeld case to thermalized conﬁgurations, was explained in Refs. [

]

by the spectral properties of the staggered Wilson kernel. While in the free-ﬁeld

limit the eigenvalue spectrum has a shape close to the Ginsparg-Wilson circle,

in the case of a

β =

6 background ﬁeld the spectrum contracts, the gap of the

spectrum becomes narrow and the branches become diﬀuse. At

β =

8, these

eﬀects become even more severe. Due to these changes in the eigenvalue spectrum

signiﬁcantly more computational time is needed, partly oﬀsetting the advantage

of a smaller fermion matrix.

At the same time we could show in Refs. [

], that at

β =

6 the staggered

Wilson kernel has a signiﬁcantly reduced condition number and is 4

–

6 times more

eﬃcient for inverting the Dirac operator on a source compared to the Wilson kernel.

The diﬀerence in the relative performance of staggered overlap vs. usual overlap

fermions compared to the one of their kernel operators as well as the connection

to spectral properties then has to be clariﬁed.

Looking at Refs. [

], we note the use of small lattices of up to a size of 12

for

benchmarking the staggered overlap operator. Generally, the use of large lattices

for overlap fermions is diﬃcult due to the high computational costs and might be

diﬃcult to justify for exploratory studies. On the other hand, the studies of the

staggered Wilson Dirac operator in Refs. [

] were carried out on lattice sizes of

up to 20

× 40.

This gives rise to the question if the observed large diﬀerences of the computa-

tional eﬃciency of overlap vs. staggered overlap fermions on small lattices com-

pared to the respective kernel operators on larger lattices are connected to changes

in the eigenvalue spectrum and chiral properties when varying the lattice size.

To answer this question, we investigate the eigenvalue spectrum of the staggered

7.2 Free-ﬁeld case

Wilson Dirac operator. We investigate how the spectrum changes when moving

from small to larger lattices and discuss how this can impact the performance of

staggered overlap fermions.

This chapter is organized as follows. In Sec. 7.2, we begin by discussing the

free-ﬁeld eigenvalue spectrum of the relevant fermion discretizations. In Sec. 7.3,

we investigate eigenvalue spectra in the case of quenched QCD background ﬁelds

and discuss how changes in the spectra aﬀect computational performance. Finally,

in Sec. 7.4 we make some concluding remarks.

7.2 Free-field case

We begin by examining the free-ﬁeld eigenvalue spectra of usual Wilson and stag-

gered Wilson fermions. Throughout this chapter, we set the lattice spacing to

a =

and, in order to avoid ambiguities, write summations explicitly.

7.2.1 Wilson fermions

Wilson fermions diﬀer from the naïve discretization by the introduction of the Wil-

son term as discussed in Subsec. 2.2.6. The Wilson term is an additional covariant

Laplacian term and we can write the resulting Dirac operator in the free-ﬁeld case

(

x, y

)

(

m + 4r

)

x, y

−







− γ



x+ ˆµ, y



+ γ



x− ˆµ, y



. (7.2.1)

Here

refers to the bare fermion mass,

r ∈

(

0, 1

]

to the Wilson parameter,

to a

4 × 4 unit matrix in spinor space, the γ

matrices to a representation of the Dirac

algebra



, γ



= δ

µ, ν

and

a,b

to the Kronecker delta. Going to momentum

space we ﬁnd

(

)

(

m + 4r

)

+ i



sin p

−r



cos p

. (7.2.2)

The eigenvalues of this matrix can be easily computed as

(

)

= m + 2r



sin





± i





sin

. (7.2.3)

7 Spectral properties

On a ﬁnite lattice the values of p

are discretized. The allowed values are

= 2π



+ ε



(7.2.4)

with

, . . . , N

−

1. In the case of periodic boundary conditions we have

0, for antiperiodic boundary conditions

2. In both cases

denotes

the number of slices in µ-direction.

7.2.2 Staggered Wilson fermions

Two-ﬂavor staggered Wilson fermions were discussed in Sec. 3.4. In the free-ﬁeld

limit, the Dirac operator has a spin ⊗ ﬂavor interpretation [42, 112] of

(

)

= m

(

⊗

)

+ i



sin p



⊗



+ r

⊗



− γ



cos p



. (7.2.5)

Explicitly multiplying out the Kronecker products we ﬁnd a 16

16 matrix, whose

eigenvalues can be calculated as

(

)

= m ± i





sin

+ r



1 ±



cos p



. (7.2.6)

Here the two “

” are to be chosen independently and the allowed range for

as for Wilson fermions, but with

, . . . , N

−

1. This expression was also

derived in Ref. [71].

For Wilson and staggered Wilson fermions, the free-ﬁeld eigenvalue spectrum

can be found in Fig. 7.2.1. As one can see, the staggered Wilson spectrum is close to

the Ginsparg-Wilson circle and we have a clear separation between the physical and

the doubler branch. Together with the fact that the staggered Wilson fermion matrix

is smaller than the Wilson fermion matrix, one expects a signiﬁcant computational

advantage for staggered overlap fermions. This was conﬁrmed by the free-ﬁeld

speedup factor of O

(

)

in the studies of Refs. [70, 71].

7.3 Quenched quantum chromodynamics

−2

−1

0 1 2 3 4 5 6 7 8

Im λ

Re λ

Figure 7.2.1:

Eigenvalue spectrum of the staggered Wilson (red) and the Wilson

Dirac operator (blue) in the free-ﬁeld case together with the Ginsparg-

Wilson circle (green).

7.3 Quenched quantum chromodynamics

Let us now consider the case of (unsmeared) quenched QCD background ﬁelds.

The calculation of eigenvalue spectra on large lattices is numerically and computa-

tionally challenging. The Dirac operator is represented by a large, nonsymmetric,

complex-valued matrix of size





, where for quantum chromodynamics

the number of colors is

3 and the number of spinor components is given by

4 for Wilson and by

1 for staggered Wilson fermions. In particular, for

larger lattices such as a

× N

32 lattice, we have more than 1

eigenvalues for the Wilson Dirac operator and more than 5

for the staggered

Wilson Dirac operator. As a result, we can compute complete eigenvalue spectra

only for small lattice sizes. For larger lattices, we have to restrict ourselves to sub-

sets of the spectrum. Of particular interest is here the physically relevant low-lying

part around the physical branch.

7.3.1 Implementation

To determine the eigenvalues of interest, our numerical implementation works

as follows. For small lattices we compute complete eigenvalue spectra using the

 [

149

] library. The library function

zgeev

computes all eigenvalues of a

7 Spectral properties

(a) 32

lattice, free ﬁeld (b) 8

lattice at β = 6 (c) 10

lattice at β = 6

Figure 7.3.1: The staggered Wilson eigenvalue spectrum at β = 6.

complex nonsymmetric matrix in double precision using the

-algorithm [

150

–

152

]. As the function requires the input matrix to be constructed explicitly (i.e.

densely) in memory, its use for larger lattices is limited by quickly growing memory

requirements. As a consequence we use this method to compute complete spectra

for lattices of up to a size of 6

for the Wilson Dirac operator and up to a size of 10

in the case of the staggered Wilson Dirac operator.

For larger lattices we use  [

153

], a library to solve large-scale eigenvalue

problems involving sparse and structured matrices. It uses the implicitly restarted

Arnoldi method [

154

] to compute the low-lying part of the eigenvalue spectrum.

Here low-lying refers to eigenvalues with smallest absolute values. The implicitly

restarted Arnoldi method is an iterative method and is not suitable for the compu-

tation of complete spectra, hence our choice of  for smaller lattices. It has

nontrivial convergence behavior, in particular choosing the number of Lanczos

basis vectors has signiﬁcant impact on the performance. The

znaupd

function

deals with the case of a complex nonsymmetric matrix in double precision. By

choosing the bare mass parameter

appropriately, one can shift the part of the

spectrum of interest to the origin. With our setup we can compute typically a few

hundred eigenvalues for lattice sizes of up to 16

32 in quenched QCD background

ﬁelds.

7.3 Quenched quantum chromodynamics

(a) 32

lattice, free ﬁeld (b) 8

lattice at β = 5.8 (c) 10

lattice at β = 5.8

Figure 7.3.2: The staggered Wilson eigenvalue spectrum at β = 5.8.

In the following, all reported numerical results are for periodic boundary condi-

tions, use the canonical choice

r =

1 and are shifted back to

m =

0. All numerical

calculations are done in double precision as the use of single precision resulted in

erroneous eigenvalues due to accumulated rounding errors.

7.3.2 Eigenvalue spectra

We now begin our discussion of the eigenvalue spectra by comparing the free-ﬁeld

case with a

β =

6 quenched QCD background ﬁeld on small lattices. In Fig. 7.3.1, we

can ﬁnd two typical examples for the spectrum for a 8

and a 10

lattice. Compared

to the free-ﬁeld spectrum in the same ﬁgure, we can see how the whole spectrum

is contracted noticeably and the gap is narrow. In Ref. [

], this degradation of

the spectrum was attributed to the four-hop terms in the staggered Wilson action,

which raise gauge ﬂuctuations to the fourth power. The physical branch and the

doubler branch get closer as the spectrum collapses into a vertical stripe and one

would expect a reduced computational eﬃciency of staggered Wilson fermions.

For

β =

8 the spectrum degrades further. In Fig. 7.3.2, we can ﬁnd two exem-

plary eigenvalue spectra. We see that the branches become even more diﬀuse and

the gap is very narrow. As a result, we expect a further decreased performance

7 Spectral properties

−0.5

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6 0.7 0.8 0.9

Im λ

Re λ

lattice, β = 6

(a) 8

lattice at β = 6

−0.5

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6 0.7 0.8 0.9

Im λ

Re λ

lattice, β = 6

(b) 10

lattice at β = 6

−0.5

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6 0.7 0.8 0.9

Im λ

Re λ

×16 lattice, β = 6

× 16 lattice at β = 6

Figure 7.3.3:

Close-up of the physical branch of the staggered Wilson Dirac operator

for increasing lattice sizes.

compared to the β = 6 case.

The situation changes when we move to larger lattices. In Fig. 7.3.3, we can ﬁnd

a close-up of the physical branch over an increasing range of lattice sizes from 8

to 12

16 at

β =

6. We can observe how for larger lattices the physical branch

becomes increasingly sharp. Finally, in Fig. 7.3.4 we can ﬁnd a direct comparison

of the physical branch of both the Wilson and staggered Wilson Dirac operator

up to the largest lattices we can access, i.e. 16

32. Here we can see very clearly

how the spectrum improves from small to large lattices. While on small lattices

the branches are diﬀuse and not well-separated, on larger lattices the situation

improves remarkably.

7.4 Conclusions

−0.2

−0.1

0.0

0.1

0.2

0.7 0.8 0.9

Im λ

Re λ

×16 lattice, β = 6

(a) 12

× 16 lattice at β = 6

−0.2

−0.1

0.0

0.1

0.2

0.7 0.8 0.9

Im λ

Re λ

×32 lattice, β = 6

(b) 12

× 32 lattice at β = 6

−0.2

−0.1

0.0

0.1

0.2

0.7 0.8 0.9

Im λ

Re λ

×32 lattice, β = 6

× 32 lattice at β = 6

Figure 7.3.4:

Comparison of the staggered Wilson (red) and Wilson Dirac spectrum

(blue).

7.4 Conclusions

Our observations give a possible explanation for the modest speedup factors for

staggered overlap fermions observed in Refs. [

] compared to the large speedup

factors for staggered Wilson fermions found in Refs. [

]. While the former study

was carried out on small lattices up to a size of 12

, the latter one was done on

lattices sizes of up to a size of 20

× 40.

As we conﬁrmed, on small lattices the eigenvalue spectrum has a small gap and

is lacking a clear separation of the branches. Here one can often ﬁnd eigenvalues

close to the center of the “belly” of the spectrum, making the overlap construc-

tion computationally more expensive. The situation quickly improves on both

the infrared and ultraviolet part of the spectrum when increasing the lattice size,

see Fig. 7.3.3. While this also holds for Wilson fermions, it appears to be more

pronounced for the staggered Wilson Dirac operator, which is likely linked to the

presence of four-hop terms in the staggered Wilson term.

That the eigenvalue spectrum of the staggered Wilson Dirac operator is well-

behaved on larger lattices can also be seen from the condition number. The diﬀuse

spectrum could in principle give rise to almost vanishing eigenvalues when the pion

mass is small. However, as we observed the condition number in our numerical

studies in Sec. 4.3 to be typically a factor of

(

)

smaller compared to Wilson

7 Spectral properties

fermions, we conclude that these pathological cases are rare.

Finally the smaller additive mass renormalization of staggered Wilson fermions

is suggesting improved chiral properties compared to Wilson fermions, which are

expected to have a positive impact on the computational eﬃciency.

As a consequence, we believe that the speedup factors reported in Refs. [

]

are not representative for the actual performance gains achievable with staggered

overlap fermions on larger lattices. We would expect a higher computational eﬃ-

ciency in cases where the staggered Wilson eigenvalue spectrum shows a wide gap

and well-separated, sharp branches. As our study suggests, this can be (at least

partially) achieved by moving to larger lattices, although only a comprehensive

numerical study can clarify this point in the end. Alternatively, as suggested in

Ref. [

], smearing also signiﬁcantly improves the eigenvalue spectrum and, thus,

is expected to result in an improved computational performance.



C 

S   

Besides staggered overlap fermions, the staggered Wilson kernel allows the con-

struction of a related chiral lattice formulation, namely staggered domain wall

fermions as proposed by Adams in Refs. [

]. In the following, we investigate

spectral properties and quantify chiral symmetry violations of various formulations

of staggered and usual domain wall fermions. We do this in the free-ﬁeld case, on

quenched thermalized background ﬁelds in the setting of the Schwinger model and

on smooth topological conﬁgurations. Moreover, we present ﬁrst results for four-

dimensional quantum chromodynamics. We closely follow the discussion from

our original

and subsequent

reports given in Refs. [

], where we previously

presented our results.

8.1 Introduction

For our understanding of the low-energy dynamics of quantum chromodynamics,

such as hadron phenomenology, chiral symmetry plays an essential role. While

the proper implementation of chiral symmetry on the lattice turned out to be a

notorious problem, it was eventually overcome with the overlap construction as

discussed in Chapter 6. Overlap fermions obey an exact chiral symmetry, thus

evading the Nielsen-Ninomiya theorem (cf. Subsec. 2.2.4). From a theoretical

perspective overlap fermions have many desirable properties, but their practical

use is limited due to the fact that they typically require a factor of

(

10–100

)

computational resources compared to Wilson fermions. Moreover, even at moder-

ate lattice spacings tunneling between topological sectors is strongly suppressed

Discussion based on and ﬁgures reprinted with permission from C. Hoelbling and C. Zielin-

ski, “Spectral properties and chiral symmetry violations of (staggered) domain wall fermions

in the Schwinger model,” Phys. Rev.

D94

no. 1, (2016) 014501,

arXiv:1602.08432 [hep-lat]

Discussion based on and ﬁgures reprinted from C. Hoelbling and C. Zielinski, “Staggered do-

main wall fermions,” PoS

LATTICE2016

(2016) 254,

arXiv:1609.05114 [hep-lat]

. Copyright

owned by the authors under the terms of the Creative Commons Attribution-NonCommercial-

NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).

8 Staggered domain wall fermions

[

155

–

158

], so that adequate coverage of the diﬀerent topological sectors becomes a

concern.

An alternative to overlap fermions is given by domain wall fermions [

159

–

161

which implement an approximate chiral symmetry in

dimensions by means of a

theory of massive coupled fermions in

d +

1 dimensions (

d =

4). The extent of

this extra dimension controls the degree of chiral symmetry violations and in the

limit of an inﬁnite extent one recovers the overlap operator with its exact chiral

symmetry. The case of a ﬁnite extent can be interpreted as a truncation of overlap

fermions.

Implementations of domain wall fermions can be easily parallelized and bear

the potential of reduced computational costs. While they only have an approxi-

mate chiral symmetry, violations of chiral symmetry are well-controlled. Based on

theoretical grounds, these violations are expected to be exponentially suppressed

[

160

162

–

164

]. In practice, however, this suppression typically still requires large

extents of the extra dimension [

165

–

169

]. At the same time, these violations allow

for an easier tunneling between diﬀerent topological sectors.

Typically domain wall fermions are formulated with a Wilson-like kernel operator.

However, recently Adams laid the foundations [

] for the use of staggered

kernels in the domain wall construction. This was enabled by the introduction of a

ﬂavored mass term as described in Sec. 3.4. While in previous studies the properties

of the staggered Wilson kernel [

–

] and the staggered overlap construction

[

] were investigated, the staggered domain wall fermion formulation was

largely ignored in the literature. For this reason, we investigate Adams’ proposal in

this chapter. We give explicit constructions of these lattice fermions in diﬀerent

variants, study their spectral properties and compare the degree of chiral symmetry

breaking with the one of usual domain wall fermions in the setting of the Schwinger

model [170].

The Schwinger model, i.e. quantum electrodynamics (QED) in two dimensions,

serves us a toy-model for quantum chromodynamics (QCD). Like quantum chro-

modynamics itself, it shows conﬁnement and has topological structure. At the

same time its numerical treatment is simpler and allows the computation of com-

plete eigenvalue spectra of lattice Dirac operators on nontrivial background ﬁelds.

Moreover, the study of the Schwinger model is of interest in its own right as it is

relevant for the description of conducting electrons in metals in the low-energy

regime, see e.g. Ref. [171].

This chapter is organized as follows. In Sec. 8.2, we give a short overview of the

8.2 Kernel operators

kernel operators we use in our study. In Sec. 8.3, we discuss the construction of

(staggered) domain wall fermions and their variants. In Sec. 8.4, we introduce

eﬀective low-energy Dirac operators and discuss their relations to the overlap

formalism. In Sec. 8.5, we introduce the setting of our numerical calculations. In

Sec. 8.6, we discuss the numerical results in detail. In Sec. 8.7, we present ﬁrst

results for four-dimensional quantum chromodynamics. In Sec. 8.8, we provide

some exemplary weight factors for the optimal domain wall construction and

ﬁnally in Sec. 8.9, we end with some concluding remarks.

8.2 Kernel operators

Throughout this chapter we consider domain wall fermions with two diﬀerent

kernel operators, i.e. Wilson and staggered Wilson fermions. In the following, we

adopt the notation previously used in Chapters 2, 3 and 6. In the case of

d =

dimensions the

∈ C

2×2

matrices (

µ =

2) correspond to a representation of

the Dirac algebra



, γ



µν

and we denote the chirality matrix as

. At

the same time, the

∈ C

2×2

matrices are a representation of the Dirac algebra in

ﬂavor space.

Wilson fermions.

We introduced and discussed the Wilson Dirac operator in Sub-

sec. 2.2.6. The Dirac operator is deﬁned in Eq.

(2.2.21)

, where we consider the case

of two space-time dimensions for our numerical studies.

Staggered Wilson fermions.

The second kernel operator of interest is Adams’ stag-

gered Wilson Dirac operator. In Sec. 3.4 and Sec. 3.6, we discussed its construction

and properties in detail. In the numerical part of this chapter we use the formu-

lation in

d =

2 dimensions as deﬁned in Eq.

(3.6.20)

. We note that in principle

also the use of the other ﬂavored mass terms in Chapter 3 can give rise to suitable

kernel operators.

8.3 Domain wall fermions

Domain wall fermion were originally proposed by Kaplan in Ref. [

159

] and subse-

quently reﬁned by Shamir and Furman in Refs. [

160

161

]. By means of a

(

d + 1

)

dimensional theory, the domain wall formulation realizes approximately massless

8 Staggered domain wall fermions

fermions in

dimensions. Alternatively, one can interpret the construction as a

tower of N

fermions in d dimensions with a speciﬁc ﬂavor structure.

We begin by introducing the

(

d + 1

)

-dimensional bulk operators, where we ﬁx

the lattice spacing to

a =

1 in the ﬁrst

dimensions and the (staggered) Wilson

parameter to

r =

1. Although for the numerical work we are interested in the

d =

2 setting, we keep the following discussion general and consider the case of

even

dimensions, where

d+1

refers to the corresponding chirality matrix in that

dimension.

8.3.1 Standard construction

We begin with the originally proposed construction. We deﬁne

= a

d+1

(

−M

)

(8.3.1)

and explicitly keep the lattice spacing

d+1

in the extra dimension. The mass

parameter

is commonly referred to as the domain wall height and has to be

suitably chosen to give rise to a one ﬂavor theory. For the free-ﬁeld case, its values

are restricted to the range M

∈

(

0, 2r

)

We can write down the Dirac operator, which reads

ΨD

Ψ =



s=1



−P

−

s+1

−P

s−1



(8.3.2)

with chiral projectors

(

± γ

d+1

)

2 and

(

d + 1

)

-dimensional fermion ﬁelds

. Throughout this chapter, the index

s =

, . . . , N

refers to the additional spatial

coordinate (or equivalently to the ﬂavor). Along the extra spatial coordinate, the

gauge links are taken to be the unit matrix. Furthermore, for ﬁnite extents of the

s-coordinate we must impose boundary conditions, which we take to be



+ mΨ



= 0,

−



+ mΨ



= 0.

(8.3.3)

Here the parameter

is related to the bare fermion mass, see Eq.

(8.4.8)

. In the

special case of

m =

0 the condition reduces to Dirichlet type boundary conditions,

while for

m = ±

1 we ﬁnd (anti-)periodic boundary conditions. Writing the Dirac

8.3 Domain wall fermions

operator explicitly in the extra dimension, we ﬁnd







−P

−

−P

−

−P

−

−P







. (8.3.4)

The d-dimensional fermion ﬁelds can be deﬁned from the boundary as

q = P

+ P

−

q = Ψ

+ Ψ

−

(8.3.5)

Now let us also introduce the reﬂection operator in the extra dimension

R =













. (8.3.6)

We can easily verify that D

is R γ

d+1

Hermitian, that is

†

= R γ

d+1

· D

· R γ

d+1

. (8.3.7)

This property implies the reality of the fermion determinant, i.e. det D

∈ R.

Besides this standard formulation, several variants of domain wall fermions have

been proposed after the original proposal.

8.3.2 Boriçi’s construction

Among them we can ﬁnd Boriçi’s construction [

172

], which follows from the stan-

dard construction by the replacements

s−1

→ −D

−

s−1

−

s+1

→ −D

−

s+1

(8.3.8)

and is an

(

d+1

)

modiﬁcation. After this replacement, the Dirac operator takes

the form

ΨD

Ψ =



s=1



+ D

−

s+1

+ D

−

s−1



(8.3.9)

8 Staggered domain wall fermions

or explicitly







−

−mD

−

−mD

−







. (8.3.10)

Moreover, the generalization of Eq. (8.3.5) reads

q = P

+ P

−

q = −Ψ

−

− Ψ

−

(8.3.11)

and the equivalent of Eq. (8.3.7) takes the form



−1



†

= R γ

d+1



−1



· R γ

d+1

D =

⊗ D

−

(8.3.12)

(see Ref. [

173

]). Boriçi referred to this formulation as truncated overlap fermions due

to the fact that the corresponding

-dimensional eﬀective operator, as discussed

in Sec. 8.4, corresponds to the polar decomposition approximation [

107

174

] of

Neuberger’s overlap operator of order N

/2 (where N

is even).

8.3.3 Optimal construction

Another modiﬁcation of domain wall fermions is the optimal construction as pro-

posed by Chiu [

175

176

]. Here one modiﬁes

so that the eﬀective Dirac operator

is formulated via Zolotarev’s optimal rational function approximation of the

sign

function [

177

–

179

], see also Refs. [

117

180

]. We now summarize the construction

given in Ref. [175].

As a starting point we use Boriçi’s formulation of domain wall fermions. We

generalize the Dirac operator by introducing weight factors

ΨD

Ψ =



s=1



(

)

+ D

−

(

)

−

s+1

+ D

−

(

)

s−1



(8.3.13)

via

(

)

= a

d+1

(

−M

)

. (8.3.14)

8.3 Domain wall fermions

The weight factors ω

take the values

min



1 − κ

(

, κ

)

(8.3.15)

with

(

, κ

)

being the respective Jacobi elliptic function with the argument

and modulus κ

. The modulus is taken as



1 − λ

min

/λ

max

(8.3.16)

and λ

min

(λ

max

) denotes the smallest (largest) eigenvalue of H

with

= γ

d+1

(

−M

)

. (8.3.17)

The argument v

reads

(

−1

)

s−1

M sn

−1





1 + 3λ

(

1 + λ

)

√

1 − λ







, (8.3.18)

where

λ =



`=1

(

2`K

, κ

)

((

2` − 1

)

, κ

)

, (8.3.19)

M =



`=1

((

2` − 1

)

, κ

)

(

2`K

, κ

)

. (8.3.20)

Here

denotes the ﬂoor function,

= K

(

)

is the complete elliptic integral of

the ﬁrst kind with

(

)

π/2



dθ

√

1 − k

sin

. (8.3.21)

We also introduced the elliptic Theta function via

(

w, k

)

= ϑ



πw

, k



(8.3.22)

with

K = K

(

)

and elliptic theta functions

, see e.g. Ref. [

181

]. To allow for the

veriﬁcation of implementations of these so-called optimal domain wall fermions,

we provide some reference values for the weight factors ω

in Sec. 8.8.

8 Staggered domain wall fermions

In the optimal construction, Eq. (8.3.5) is replaced by

q = P

+ P

−

q = −Ψ

−

(

)

− Ψ

−

(

)

−

(8.3.23)

and Eq. (8.3.7) again takes the form of Eq. (8.3.12), but now with

D = diag

[

−

(

)

, . . . , D

−

(

)]

(8.3.24)

as pointed out in Ref. [

173

]. Optimal domain wall fermions are one of the most

popular domain wall fermion formulations and were used in numerous studies. In

Refs. [182–187] some of these works can be found.

Besides the original optimal domain wall fermion formulation, there is also a

modiﬁed version [

188

] which is reﬂection-symmetric along the extra dimension.

Finally, we note that all of the preceding domain wall fermion variants can be

interpreted as special cases of Möbius domain wall fermions [173, 189, 190].

8.3.4 Staggered formulations

As recently clariﬁed by Adams, it is possible to employ a staggered kernel for the for-

mulation of domain wall fermions [

]. Following this idea, the Dirac operator

takes the form

ΥD

sdw

Υ =



s=1



−P

−

s+1

−P

s−1



, (8.3.25)

where

refers to the

(

d + 1

)

-dimensional staggered fermion ﬁeld. Similarly to the

Wilson case, we let

= a

d+1

(

−M

)

(8.3.26)

with

being the staggered Wilson Dirac operator. The chiral projectors are given

(

± 

)

2, where



is deﬁned in Eq.

(3.6.18)

and we note that



. Recall

that

 ∼ γ

d+1

⊗ ξ

d+1

, which reduces to

 ∼ γ

d+1

⊗

on the physical species. The

Rγ

d+1

Hermiticity of

generalizes to a

R

Hermiticity of

sdw

. With our sign

convention

sdw

is in full analogy with

, while a slightly diﬀerent convention

was used in the original proposal in Ref. [45].

In Ref. [

], a replacement rule was speciﬁed, which can be written in our general

8.4 Eﬀective Dirac operator

d-dimensional setting as

d+1

→ , D

→ D

(8.3.27)

and describes how

sdw

can be constructed from

. Using Eq.

(8.3.27)

, it is

possible for us to generalize also Boriçi’s and Chiu’s work to the case of a staggered

Wilson kernel. This allows us to write down the Dirac operator for what we are

going to denote as truncated staggered domain wall fermions

and which reads

ΥD

sdw

Υ =



s=1



+ D

−

s+1

+ D

−

s−1



. (8.3.28)

Moreover, the Dirac operator of optimal staggered domain wall fermions, a gener-

alization of Chiu’s construction, takes the form

ΥD

sdw

Υ =



s=1



(

)

+ D

−

(

)

−

s+1

+ D

−

(

)

s−1



. (8.3.29)

Here we let

(

)

= a

d+1

(

−M

)

and the weight factors

are given by

Eq. (8.3.15) for the kernel H

= D

(

−M

)

8.4 Effective Dirac operator

We now discuss the eﬀective low-energy

-dimensional Dirac operator derived

in Refs. [

172

191

–

193

] (cf. Refs. [

194

195

]). This will shine some light on the rela-

tion between the light

-dimensional

ﬁelds at the boundary and the

(

d + 1

)

dimensional theory. We follow Refs. [

172

192

195

] in our brief discussion of the

construction.

8.4.1 Derivation

After integrating out the

−

1 heavy modes in the theory, one can deﬁne an

eﬀective low-energy d-dimensional action

eff



(

)

eff

(

)

, (8.4.1)

We chose the name “truncated staggered domain wall fermions” in Refs. [

] rather than “trun-

cated staggered overlap fermions” in order to emphasize the fact that we are dealing with a

(

d + 1

)

-dimensional domain wall fermion formulation, which is motivated by a truncation of the

staggered overlap operator.

8 Staggered domain wall fermions

where the eﬀective Dirac operator is deﬁned by means of the propagator of light

modes

−1

eff

(

x, y

)

(

)

(

. (8.4.2)

In the

(

d + 1

)

-dimensional theory we ﬁnd one light and

−

1 heavy Dirac fermions

(for suitable choices of M

If we take the chiral limit, where at ﬁxed bare coupling

we take the extent of

the extra dimension to inﬁnity

→ ∞

, the contribution from the heavy modes

diverges. This bulk contribution can be canceled by the introduction of pseud-

ofermionic ﬁelds, where the pseudofermion action is typically taken to be the

fermion action with m = 1.

Now let us deﬁne the Hermitian operators

= γ

d+1

(

−M

)

, H

= γ

d+1

(

−M

)

, (8.4.3)

where for standard domain wall fermions the kernel operator reads

(

−M

)

(

−M

)

2 ·

+ a

d+1

(

−M

)

. (8.4.4)

The transfer matrix in the extra dimension takes the form

T =

−

, T

± a

d+1

H, (8.4.5)

where the Hermitian operator H depends on the construction given by

H =











for standard construction,

for Boriçi’s construction.

(8.4.6)

One can derive the explicit form [192] of the eﬀective Dirac operator to ﬁnd

eff

1 + m

1 − m

d+1

−T

−

+ T

−

. (8.4.7)

Rewriting Eq. (8.4.7) as

eff

(

1 − m

)



eff

(

)

1 − m



, (8.4.8)

where

eff

(

)

refers to the eﬀective operator

eff

m =

0, we ﬁnd a bare mass of

(

1 − m

)

for a given choice of the parameter m, see Ref. [196].

8.4 Eﬀective Dirac operator

The eﬀective operator in Eq. (8.4.7) can also be written in the compact form

eff



−1



1, 1

(8.4.9)

as shown in Refs. [172, 195], where the matrix P reads

P =







−







(8.4.10)

and

−1

= P

. To simplify the notation we introduced

≡ D

(

)

and the

(

1, 1

)

-index refers to the corresponding s -block of the matrix product.

For Chiu’s optimal domain wall fermions the derivation follows Boriçi’s case,

where the additional weight factors

have to be taken into account [

175

]. By con-

struction, we ﬁnd that in this case the

sign

-function approximation in the eﬀective

Dirac operator equals Zolotarev’s optimal rational function approximation.

8.4.2 The N

→ ∞ limit

For the following discussion we restrict ourselves to the

m =

0 case. First, we

remark that Eq. (8.4.7) can be written as

−T

−

+ T

−

= 

(

d+1

)

, (8.4.11)

where



is Neuberger’s polar decomposition approximation [

107

174

] of the

sign

-function. One can easily see that in the

→ ∞

limit we recover the overlap

Dirac operator,

= lim

→∞

eff

d+1

sign H



+ D

−M



†

−M



−



, (8.4.12)

8 Staggered domain wall fermions

with H given in Eq. (8.4.6), using the shorthand notation D

−M

= D

(

−M

)

and

D =











for standard construction,

for Boriçi’s/Chiu’s construction.

(8.4.13)

The Ginsparg-Wilson equation

{

d+1

, D

}

= 2D

d+1

(8.4.14)

can be shown to hold for

, thus implementing an exact chiral symmetry and

implying the normality of the operator. We note that compared to the discussion

in Subsec. 6.1.1, we ﬁnd an additional factor of two in the Ginsparg-Wilson relation

due to a diﬀerent normalization of the overlap operator.

If we now compare Eq. (8.4.12) with the standard deﬁnition

= ρ



+ D

−ρ



†

−ρ



−



(8.4.15)

of the overlap operator and we use the relation for the eﬀective negative mass

parameter

ρ =











−

d+1

for standard construction,

for Boriçi’s/Chiu’s construction,

(8.4.16)

this would result in a restriction on the domain wall height

from

ρ =

2. This

limitation can be overcome by rescaling

eff

by a factor

% =

, so that in the

free-ﬁeld case the low-lying eigenvalues of the kernel operator remain invariant

(up to discretization eﬀects) when the eﬀective operator projection is applied. This

can also be seen in Fig. 8.6.7, which we later discuss in Sec. 8.6.1. In all numerical

investigations of this chapter this rescaling is taken into account.

8.4.3 Approximate sign functions

The eﬀective operator, as previously derived, is an approximation to the overlap

operator. Explicitly, the sign-function is approximated by the rational function

(

)

(

)

− Π

−

(

)

(

)

+ Π

−

(

)

(8.4.17)

8.5 Setting

−1.0

−0.5

0.0

0.5

1.0

−4.0 −3.0 −2.0 −1.0 0.0 1.0 2.0 3.0 4.0

Polar

Optimal

(a) N

= 2

0.99996

0.99998

1.00000

1.00002

1.00004

1.0 1.5 2.0 2.5 3.0

Polar

Optimal

(b) N

= 6

Figure 8.4.1:

Approximations of the

sign

-function by

(

)

. The optimal construc-

tion is illustrated for the case of λ

min

= 1 and λ

max

= 3.

with

(

)











(

1 ± z

)

for standard/Boriçi’s construction,



(

1 ± ω

)

for Chiu’s construction.

(8.4.18)

As expected, we can verify that

(

)

→ sign

(

)

for

→ ∞

. We note that for the

standard construction and Boriçi’s construction the approximation is identical,

but is applied to diﬀerent kernel operators, namely

respectively. To il-

lustrate the approximations in Eq.

(8.4.18)

, we compare the polar decomposition

approximation in Boriçi’s formulation with the optimal approximation in Chiu’s

construction in Fig. 8.4.1. As we can see, the approximation error over the construc-

tion range is signiﬁcantly lower for the optimal construction. We recall that the

coeﬃcients ω

are related to Zolotarev’s coeﬃcients, cf. Refs. [117, 177, 180].

Staggered formulations

The discussion in this section naturally generalizes to the case of a staggered Wilson

kernel, when the replacement rule in Eq.

(8.3.27)

is applied. In particular, the

eﬀective and overlap operators follow from this replacement and we can interpret

staggered domain wall fermions as a truncation of staggered overlap fermions [

8.5 Setting

We now move on to the numerical part of the study and begin by discussing our

setting. In particular, we elaborate on our method to quantify chiral symmetry

violations to benchmark the various domain wall fermion constructions.

8 Staggered domain wall fermions

8.5.1 Setting the domain wall height

To give rise to a one-ﬂavor theory, in the free-ﬁeld case the domain wall height

has to be chosen in the range 0

< M

. When moving to a nontrivial

background ﬁeld, this interval generally shifts and contracts. In less rigor terms,

the parameter

has to be chosen so that the origin is shifted inside the leftmost

“belly” of the eigenvalue spectrum of the kernel operator. Although even in the

free-ﬁeld case there is no optimal choice for

as discussed in Ref. [

197

], one

typically uses the canonical choice

1 for the free ﬁeld, which is at the center

of the eigenvalue belly. More rigorously speaking, valid choices of

are restricted

by the position of the mobility edge [51, 198–200].

In the Schwinger model, for reasonable choices of the inverse coupling

the

eigenvalue spectrum is not distorted too far away from the free-ﬁeld spectrum.

As a consequence, valid choices for

remain close to the free-ﬁeld case. While

in four-dimensional quantum chromodynamics one often uses

8 (see e.g.

Ref. [

201

]), in our present case

1 remains a sensible and simple choice, which

we consequently use for all our numerical work.

8.5.2 Effective mass

Commonly employed measures in the literature to quantify the degree of chiral

symmetry breaking of domain wall fermions are the residual mass

res

[

161

167

201

202

] and the eﬀective mass

eff

[

203

–

205

]. While the residual mass makes use of

the explicit mass dependence in the chiral Ward-Takahashi identities, the eﬀective

mass is deﬁned as the lowest eigenvalue (by magnitude) of the Hermitian operator

in a background ﬁeld with nontrivial topology. Although these measures are not

identical, they usually agree within a factor of order

(

)

. Consequently both deﬁne

valid measures to quantify the chirality of a given lattice fermion formulation.

Because of its conceptual simplicity, we use the eﬀective mass

eff

as one of

our measures for the numerical studies. To give a precise deﬁnition, we consider

a given Dirac operator

on a topologically nontrivial background conﬁguration

with periodic boundary conditions and deﬁne the eﬀective mass

eff

as the lowest

eigenvalue of the Hermitian kernel operator H . Explicitly, we let

eff

= min

λ∈spec H

= min

Λ∈spec D

†

√

Λ, (8.5.1)

where we used the fact that

= D

†

. For a normal operator

, this deﬁnition

8.5 Setting

simpliﬁes to

eff

= min

λ∈spec D

. For a non-normal operator, however, the

eigenvalues of D and H are not directly related.

For gauge conﬁgurations with topological charge

Q ,

0, the existence of zero

modes of

is guaranteed in the continuum by the Atiyah-Singer index theorem

[

121

–

124

]. Similarly, the presence of zero modes of the overlap operator as deﬁned

in Eq.

(8.4.12)

can be shown as well [

109

–

111

]. For the eﬀective Dirac operators these

zero modes are approximate, but become exact in the

→ ∞

limit. The absolute

value of the eigenvalues of these approximate zero modes can then serve as a

measure of chirality.

In the context of a gauge theory, the Atiyah-Singer index theorem takes the

following form. Let

∓

refer to the number of left-handed and right-handed zero

modes, then their diﬀerence is related to the topological charge by

−

− n

(

−1

)

d/2

Q, (8.5.2)

see Ref. [

]. In Eq.

(8.5.6)

, we will give a precise deﬁnition of

. We note that one

can show the Vanishing Theorem [

125

–

127

] in our two-dimensional setting. The

theorem states that on gauge conﬁgurations with

Q ,

0, either

−

vanishes,

i.e. n

−

and n

are not simultaneously nonzero.

8.5.3 Normality and Ginsparg-Wilson relation

The continuum Dirac operator, the naïve and overlap lattice Dirac operators are all

normal operators. At the same time, the eﬀective Dirac operators are not normal.

This makes operator normality an interesting aspect in the context of chirality,

especially as it has been shown that chiral properties imply the normality of the

Dirac operator [206], see also Refs. [207, 208].

By deﬁnition a normal operator

satisﬁes



D, D

†



0. Deviation from normal-

ity can be quantiﬁed by the norm of this commutator and we deﬁne

∆





D, D

†





∞

, (8.5.3)

where

∞

is the matrix norm induced by the

∞

-norm. As previously discussed,

the normality measure

∆

vanishes for the overlap operator and, thus, for the

eﬀective Dirac operators in the limit N

→ ∞.

Besides normality, violations of the Ginsparg-Wilson relation as given in Eq.

(8.4.14)

8 Staggered domain wall fermions

are a natural measure for chirality of a lattice fermion formulation. We let

∆



{

, D

}

− ρ

−1

Dγ



∞

(8.5.4)

and ﬁnd that for the eﬀective operators

∆

→

0 for

→ ∞

. For a staggered

Wilson kernel, the chirality matrix

has to be replaced by the staggered



in the

deﬁnition of

∆

. We note that the additional factor of 1

/ρ

in Eq.

(8.5.4)

is due

to the diﬀerent scales of the eﬀective Dirac operators. The measures

∆

and

∆

were previously considered in Refs. [

197

209

] and together with

eff

serve here as

a measure for chiral symmetry violations.

8.5.4 Topological charge

To determine the topological charge of a gauge conﬁguration, we use the index

formula for the standard overlap operator,

Q =







, (8.5.5)

and its staggered equivalent,

Q =







, (8.5.6)

with

= D

(

−M

)

as discussed in Ref. [

]. We numerically veriﬁed that these

two deﬁnition yield the same topological charge on a set of sample conﬁgurations.

This observation is in agreement with analytical results [

] and other numerical

studies [71, 210].

8.6 Numerical results

For the numerical part of this work, we calculate complete eigenvalue spectra and

evaluate our measures of chirality for the previously discussed Dirac operators. We

do this on various gauge conﬁgurations in the Schwinger model and cover the free-

ﬁeld case, thermalized conﬁgurations and the smooth topological conﬁgurations

discussed in Ref. [211].

Throughout this section, we set the lattice spacings in all dimensions to

a =

d+1

1 and ﬁx the (staggered) Wilson parameter to

r =

1. For the extra dimension

we consider extents within the range 2

≤ N

≤

8. In order to allow the calculation

100

8.6 Numerical results

of the eﬀective mass

eff

as discussed in Sec. 8.5.2, we impose periodic boundary

conditions in all dimensions.

We determine complete eigenvalue spectra using the  [

149

] library, while

extremal eigenvalues, such as the eﬀective mass, are computed with  [

153

For all numerical calculations we use double precision. In the ﬁgures, the standard

construction is abbreviated with “std”, Boriçi’s construction with “Bor” and Chiu’s

optimal construction with “opt”. When discussing overlap operators, the construc-

tion with kernel

is referred to as “DW”, Neuberger’s overlap with kernel

abbreviated as “Neub” and Adams’ staggered overlap with kernel

is denoted

as “Adams”.

8.6.1 Free-field case

In our discussion we begin with the simplest case of the free ﬁeld. Periodicity allows

a Fourier transformation of the kernel operators and simpliﬁes the computation

of the eigenvalue spectra, cf. Chapter 7. In the momentum representation, the

Wilson kernel takes the form of the following 2 × 2 linear map

(

+ 2r

)

+ i



sin p

−r



cos p

, (8.6.1)

where

πn

with

, . . . , N

−

1 and

denotes the number of

slices in

-direction. The staggered Wilson kernel is represented by the 4

4 linear

map

= m

(

⊗

)

+ i



sin p



⊗



+ r

⊗



+ ξ



cos p



(8.6.2)

with n

= 0, 1, . . . , N

/2 − 1, cf. Ref. [71].

For the three-dimensional bulk operator, the extra dimension is lacking periodic-

ity and we leave the corresponding coordinate in the position-space representation.

By employing a momentum-space representation for the kernel operators, we re-

duce the dimensionality of the problem and avoid numerical instabilities otherwise

encountered in the free-ﬁeld case. All free-ﬁeld results discussed in the following

are for the case of a N

× N

= 20 × 20 lattice.

Kernel operators.

In Fig. 8.6.1, we show the free-ﬁeld eigenvalue spectra of the Wil-

son and staggered Wilson Dirac operator. While in four dimensions Wilson fermi-

101

8 Staggered domain wall fermions

0.5

1.5

2.5

3.5

Re(λ)

-1.5

-1

-0.5

0.5

1.5

Im(λ)

Wilson

0.5

1.5

2.5

3.5

Re(λ)

-1.5

-1

-0.5

0.5

1.5

Im(λ)

staggered Wilson

Figure 8.6.1: Spectrum of the kernel operators in the free ﬁeld case.

ons have four doubler branches, in our setting of the two-dimensional Schwinger

model only two doubler branches are present. Adams’ staggered Wilson term,

however, always splits the tastes into two groups with positive and negative ﬂavor-

chirality. Compared to Wilson fermions, we observe how the spectrum of the

staggered Wilson kernel resembles the Ginsparg-Wilson circle much more closely.

On suﬃciently smooth conﬁgurations one can then hope for lower computational

costs for the construction of chiral fermion formulations.

We note that the Wilson and staggered Wilson eigenvalue spectrum diﬀers mostly

in their ultraviolet parts and very little in the physically relevant low-lying part.

Nevertheless, these discretization eﬀects alter the properties and shape of the bulk,

eﬀective and overlap operators and in general have an impact on computational

eﬃciency and chiral properties.

Bulk operators.

Moving on to the

(

2 + 1

)

-dimensional bulk operators, in Figs. 8.6.2

and 8.6.3 we show the eigenvalue spectra in the standard, Boriçi’s and Chiu’s con-

struction. To illustrate the eﬀect of changing boundary conditions in the extra

dimension, we show in Fig. 8.6.4 the case of periodic (

m = −

1) and antiperiodic

(

m =

1) boundary conditions, which can be compared to the Dirichlet (

m =

)

case.

In the spectrum of the bulk operator with the standard construction, we ﬁnd

(two) three doubler branches for the case of a (staggered) Wilson kernel. For Boriçi’s

102

8.6 Numerical results

-2 0 2 4

Re(λ)

-2

Im(λ)

(a) Standard construction

-2 0 2 4

Re(λ)

-2

Im(λ)

(b) Boriçi’s construction

-2 0 2 4

Re(λ)

-2

Im(λ)

Figure 8.6.2:

Spectrum of

with Wilson kernel for

m =

0 at

8 in the free

ﬁeld case.

-2 0 2 4

Re(λ)

-2

Im(λ)

(a) Standard construction

-2 0 2 4

Re(λ)

-2

Im(λ)

(b) Boriçi’s construction

-2 0 2 4

Re(λ)

-2

Im(λ)

Figure 8.6.3:

Spectrum of

sdw

with staggered Wilson kernel for

m =

0 at

8 in

the free ﬁeld case.

construction the number of branches is reduced by one. In both cases we ﬁnd 2

exact zero modes in the Dirichlet case [

160

203

], which disappear for non-vanishing

values of

. For Chiu’s optimal construction we only ﬁnd two approximate zero

modes and observe that the real eigenvalues are spread along the real axis. In the

case of Boriçi’s and the optimal construction with a Wilson kernel, we notice how

the spectrum of the bulk operators lost its resemblance to the three-dimensional

Wilson operator.

Effective operators.

In Figs. 8.6.5 and 8.6.6, we can ﬁnd the eigenvalue spectra

of the eﬀective Dirac operators

eff

with diﬀerent kernel operators as deﬁned

in Eq.

(8.4.7)

. The spectra resemble the Ginsparg-Wilson circle and rapidly con-

verge towards it for increasing values of

. As the free ﬁeld is smooth, this fast

convergence is not unexpected. Already for small values of

(such as

8),

it is almost impossible to visually distinguish the spectrum from the one of the

associated overlap operator. We also point out the rapid convergence of Boriçi’s

and Chiu’s construction, in particular with a staggered Wilson kernel.

103

8 Staggered domain wall fermions

-2 0 2 4

Re(λ)

-2

Im(λ)

(a) Periodic case (m = −1)

-2 0 2 4

Re(λ)

-2

Im(λ)

(b) Antiperiodic case (m = 1)

Figure 8.6.4:

Spectrum of

with Wilson kernel in the standard construction at

8 in the free ﬁeld case with diﬀerent boundary conditions (cf.

Fig. 8.6.2)

By construction, the optimal construction shows signiﬁcantly improved chiral

properties. For a given spectral range

I =

[

min

, λ

max

]

of the Hermitian kernel op-

erator, the optimal rational function approximation

opt

(

)

minimizes the maximal

deviation from the sign function

max

= max

z ∈−I ∪I



sign

(

)

−r

opt

(

)



= 1 ∓r

opt

(

±λ

min

)

. (8.6.3)

For a normal operator, such as in the free-ﬁeld case, the eigenvalues are conﬁned in

a tube around the Ginsparg-Wilson circle with radius

max

. As Zolotarev’s optimal

approximation to the

sign

-function minimizes the

∞

-norm, we ﬁnd a point of

maximal deviation at both

min

and

max

, resulting in the absence of an exact zero

mode. However, this approximate zero mode is decreasing rapidly in magnitude

with increasing values of

due to the rapid convergence of the underlying

sign

function approximation.

Overlap operators.

Taking the

→ ∞

limit of the eﬀective Dirac operators, we

ﬁnd an overlap operator. The Hermitian kernel operator

of the overlap operator

as given in Eq.

(8.4.6)

depends on the choice of the domain wall construction. In

Fig. 8.6.7, we show the eigenvalue spectrum of

together with the stereographic

projection π of the eigenvalues, given by the map

(

)

1 −

. (8.6.4)

104

8.6 Numerical results

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

std Bor

opt

(a) Wilson kernel

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

std Bor

opt

(b) Staggered Wilson kernel

Figure 8.6.5:

Spectrum of

eff

for the standard (std), Boriçi (Bor) and optimal (opt)

construction at N

= 2.

The result is the projection of the eigenvalues onto the imaginary axis, where the

physical low-lying eigenvalues line up as required due to the introduction of the

scale factor in Eq. (8.4.16).

For Adams’ overlap operator the eigenvalue spectrum is highly symmetric. By

construction, the eﬀective operators in Boriçi’s and the optimal construction con-

verge, depending on the kernel operator, towards Neuberger’s and Adams’ overlap

operator for

→ ∞

. For the standard construction we ﬁnd an overlap operator

with a modiﬁed kernel operator.

8.6.2 U

(

)

gauge field case

After the discussion of the free-ﬁeld case, we now consider nontrivial gauge conﬁg-

urations. The Schwinger model is an Abelian gauge theory with symmetry group

(

)

and in the following we consider quenched thermalized gauge conﬁgurations

with the setup of Refs. [

212

213

]. Although we restrict ourselves to quenched con-

ﬁgurations, it is in principle possible to reweight to an unquenched ensemble

with arbitrary mass [

212

–

215

]. We begin our discussion with the analysis of the

eigenvalue spectra on a few selected 20

conﬁgurations at an inverse coupling of

β = 5.

105

8 Staggered domain wall fermions

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

std Bor

opt

(a) Wilson kernel

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

std Bor

opt

(b) Staggered Wilson kernel

Figure 8.6.6:

Spectrum of

eff

for the standard (std), Boriçi (Bor) and optimal (opt)

construction at N

= 8.

Kernel operators.

An example of a kernel eigenvalue spectrum on a gauge con-

ﬁguration with topological charge

Q =

1 can be found in Fig. 8.6.8. Compared to

the case of four-dimensional quantum chromodynamics, the branches are much

sharper and better separated, cf. Chapter 7 and Refs. [4, 7, 8, 70, 71, 82].

Bulk operators.

For the eigenvalue spectra of the bulk operators in Figs. 8.6.9 and

8.6.10 we use the same gauge background ﬁeld as in Fig. 8.6.8. Being a topologically

nontrivial conﬁguration, the eﬀective operator has

approximate zero modes

which become exact in the limit

→ ∞

. For the bulk operators, there are

eigenvalues in the neighborhood of the origin. It is an interesting observation to

see how the eigenvalue spectrum is distorted in the optimal construction in order

to improve chiral properties.

Effective operators.

Again using the same gauge conﬁguration, in Figs. 8.6.11 and

8.6.12 we ﬁnd the eigenvalue spectra of the eﬀective operators. Here Boriçi’s con-

struction shows superior chiral properties compared to the standard construction

for N

≥ 4 with respect to our measures m

eff

, ∆

and ∆

The optimal construction further improves upon that and shows an ever lower

degree of chiral symmetry violations. In the case of Fig. 8.6.12b, the eﬀective mass

eff

is already so small that it is in the order of the round-oﬀ error and, thus, we are

only able to quote an upper bound. In this nontrivial background ﬁeld the kernel

106

8.6 Numerical results

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

DW Neub

(a) Wilson kernel

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

DW Adams

(b) Staggered Wilson kernel

Figure 8.6.7:

Spectrum of

with its stereographic projection for the domain

wall (DW) kernel and the standard overlap (Neub/Adams) kernel.

operators lose their normality and the maximum deviation

max

of eigenvalues

from the Ginsparg-Wilson circle in the optimal construction does not have to be

saturated. We also note that a smaller

max

does not necessarily imply a smaller

value of

∆

, but that also the distribution of eigenvalues of

is important. There

is, however, a strong correlation between larger

and smaller values of

∆

, where

the violations of the Ginsparg-Wilson relation rapidly tend to zero for N

→ ∞.

Chiu’s construction is only in a very particular sense optimal, namely the min-

imization of the deviation

max

from the

sign

-function as deﬁned in Eq.

(8.6.3)

With respect to other criteria, such as the number of iterations required for solving

a linear system, optimal domain wall fermions are typically not optimal [

189

]. In

theory, it is possible to formulate variants of domain wall fermions which are opti-

mal with respect to other measures such as

∆

. However, such a construction

would likely require more knowledge about the eigenvalue spectrum of H .

For the standard construction

eff

∆

and

∆

have typically the same order of

magnitude for both a Wilson and a staggered Wilson kernel. On the other hand, for

Boriçi’s and the optimal construction a staggered Wilson kernel results in reduced

chiral symmetry violations compared to the Wilson kernel.

An interesting, although rather artiﬁcial case is

2. Here we observe that the

performance of all constructions is comparable in the case of a staggered Wilson

kernel, while for the case of a Wilson kernel the standard construction performs

107

8 Staggered domain wall fermions

0.5

1.5

2.5

3.5

Re(λ)

-1.5

-1

-0.5

0.5

1.5

Im(λ)

7.85×10

-2

2.23×10

9.31×10

8.73×10

-2

1.78×10

2.62×10

eff

∆

Wil

stag

Figure 8.6.8: Spectrum of kernel operators in a U

(

)

background ﬁeld.

better than Boriçi’s, which itself clearly outperforms the optimal construction.

Overlap operators.

In Fig. 8.6.13, we again show the spectrum of the associated

overlap operators together with a stereographic projection of their eigenvalues. We

omit the chirality measures

eff

∆

and

∆

in the ﬁgure labels as they vanish up

to round-oﬀ errors.

8.6.3 Smearing

Smearing is a common technique in the simulations of lattice gauge theories. It is

expected to be especially beneﬁcial for the staggered Wilson kernel (in particular in

1 dimensions) as discussed in Ref. [

]. For this reason we consider gauge ﬁelds

with three-step  smearing [

216

217

]. For the smearing parameter, we choose

the maximal value

α =

5 in two dimensions within the perturbatively reasonable

range [218].

To test the eﬀects of smearing on the violations of chiral symmetry, in Figs. 8.6.14

and 8.6.15 we directly compare the eﬀective operators on an unsmeared and smeared

version of a gauge conﬁguration at

4. We observe that after three smearing

steps, chiral symmetry violations as measured by

eff

∆

and

∆

are reduced

108

8.6 Numerical results

-2 0 2 4

Re(λ)

-4

-2

Im(λ)

(a) Standard construction

-2 0 2 4

Re(λ)

-4

-2

Im(λ)

(b) Boriçi’s construction

-2 0 2 4

Re(λ)

-4

-2

Im(λ)

Figure 8.6.9: Spectrum of D

with Wilson kernel for m = 0 at N

= 8.

-2 0 2 4

Re(λ)

-4

-2

Im(λ)

(a) Standard construction

-2 0 2 4

Re(λ)

-4

-2

Im(λ)

(b) Boriçi’s construction

-2 0 2 4

Re(λ)

-4

-2

Im(λ)

Figure 8.6.10: Spectrum of

sdw

with staggered Wilson kernel for

m =

0 at

signiﬁcantly. In some cases we observe a reduction by up to two order of magni-

tude with the exception of the eﬀective mass for the standard construction with a

Wilson kernel.

8.6.4 Topology

On topologically nontrivial conﬁgurations with

Q ,

0, the Atiyah-Singer index

theorem ensures that the continuum Dirac operator has exact zero modes. On the

lattice the same can be shown for the overlap operator, while for the eﬀective Dirac

operators we only ﬁnd approximate zero modes as illustrated in Fig. 8.6.16. These

modes come with a multiplicity of

as expected from the Vanishing Theorem.

By using the technique outlined in Ref. [

211

] for constructing a smooth conﬁgura-

tion for any given topological charge

, we can study topological aspects also in a

more abstract setting. The gauge conﬁgurations are maximally smooth in the sense

that they are invariant under the application of the  smearing prescription. We

109

8 Staggered domain wall fermions

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

1.75×10

-2

2.38×10

-1

8.05×10

-1

1.95×10

-2

8.84×10

-1

1.44×10

1.12×10

-1

1.43×10

1.80×10

eff

∆

std

Bor

opt

(a) Wilson kernel

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

2.01×10

-2

2.40×10

-1

7.74×10

-1

2.19×10

-2

6.68×10

-1

7.04×10

-1

4.57×10

-2

5.59×10

-1

5.55×10

-1

eff

∆

std

Bor

opt

(b) Staggered Wilson kernel

Figure 8.6.11:

Spectrum of

eff

for the standard (std), Boriçi (Bor) and optimal

(opt) construction at N

= 2.

measure

eff

∆

and

∆

for the eﬀective Dirac operators on a wide range of

smooth topological conﬁgurations with various

. Our measures of chirality are

identical for the conﬁgurations with ±Q and only depend on

Due to the smoothness of the conﬁgurations, the measures

eff

∆

and

∆

are all small compared to the typical values one measures on thermalized conﬁg-

urations at reasonable values of

. We also observe that with increasing values

the chiral symmetry violations become larger. Finally, we show two eigen-

value spectra of bulk operators on a 20

lattice in Fig. 8.6.17, revealing a very clear

structure of the spectrum on a topologically nontrivial conﬁguration.

8.6.5 Spectral flow

Besides the study of the eigenvalue spectra, topological aspects—such as the index

theorem—can be studied on the lattice also with the help of spectral ﬂows [

105

219

Commonly used with a Wilson kernel, one studies the eigenvalues

(

)

the Hermitian operator

(

)

as a function of

. One ﬁnds a direct one-to-

one correspondence between the eigenvalue crossings of

(

)

and the real

eigenvalues of

(

)

. Moreover, the low-lying real eigenvalues of the Wilson

110

8.6 Numerical results

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

2.00×10

-4

4.05×10

-2

1.08×10

-1

4.74×10

-6

5.39×10

-3

1.92×10

-2

3.36×10

-6

2.57×10

-4

3.39×10

-4

eff

∆

std

Bor

opt

(a) Wilson kernel

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

2.82×10

-4

3.26×10

-2

9.12×10

-2

3.55×10

-6

1.03×10

-3

1.01×10

-3

×10

-8

5.03×10

-6

5.71×10

-6

eff

std

Bor

opt

(b) Staggered Wilson kernel

Figure 8.6.12:

Spectrum of

eff

for the standard (std), Boriçi (Bor) and optimal

(opt) construction at N

= 8.

Dirac operator

(

)

correspond to the would-be zero modes [

211

]. For each

respective mode, the slope of the eigenvalue crossings at small values of

equals

minus its chirality. By studying the eigenvalue crossings of the Hermitian operator

in a background ﬁeld we can, thus, determine the respective topological charge.

Although originally spectral ﬂows of Wilson-like kernels were studied, Adams’

generalized the method to staggered kernels [

] allowing the study of topological

aspects within the staggered framework [

210

220

]. While in the literature

these spectral ﬂows have been investigated before, here we want to show the

eﬀectiveness of smearing when studying topological aspects on the lattice.

To this end we show an example for the ﬂow

(

)

of the lowest 50 eigenval-

ues for the Wilson kernel

(

)

and the staggered Wilson kernel

(

)

Figs. 8.6.18 and 8.6.19. The 12

gauge conﬁguration under consideration was gener-

ated at

β =

8 and has a topological charge of

Q =

2. We compare the eigenvalue

ﬂow on both the unsmeared and the three-step  smeared (

α =

5) version

of the conﬁguration and also added the smooth

Q =

2 topological conﬁguration

described in Sec. 8.6.4. While the determination of the topological charge

via the

spectral ﬂow is not easily possible for the rough and unsmeared conﬁguration, the

use of smearing allows an unambiguous identiﬁcation of the topological sector. In

111

8 Staggered domain wall fermions

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

DW Neub

(a) Wilson kernel

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

DW Adams

(b) Staggered Wilson kernel

Figure 8.6.13:

Spectrum of

with its stereographic projection for the domain

wall (DW) kernel and the standard overlap (Neub/Adams) kernel.

the case of the smooth topological conﬁguration, the two eigenvalue crossings lie

essentially on top of each other. The direct comparison between the ﬁgures shows

the eﬀectiveness of smearing for the study of topological aspects on the lattice.

8.6.6 Approaching the continuum

To evaluate the chiral properties of the various eﬀective Dirac operators when

approaching the continuum, we evaluated them on seven ensembles with 1000

conﬁgurations each. While keeping the physical volume ﬁxed, we consider the

following ensembles: 8

β =

8, 12

β =

8, 16

β =

2, 20

β =

0, 24

β =

2, 28

β =

8 and 32

β =

8. Besides the unsmeared conﬁgurations,

we also consider the smeared versions, resulting in

N =

14 000 gauge conﬁgurations

in total.

While we are interested in the comparison of the diﬀerent formulations on ﬁner

lattices, we do not attempt a rigorous continuum limit analysis. The chirality

measures on our ﬁnest lattice at

β =

8 serve us as an indicator for the relative

performance of the diﬀerent domain wall formulations. In Table 8.6.1, we quote

the median values for the parameter range

∈

{

2, 4, 6

}

. For

≥

8 some values

already become so small, that they are comparable with the round-oﬀ error and,

thus, cannot be considered reliable. For the standard construction we ﬁnd that

the chiral symmetry violations at large values of

are comparable for the cases of

112

8.6 Numerical results

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

3.04×10

-3

1.72×10

-1

5.11×10

-1

6.87×10

-4

1.42×10

-1

3.30×10

-1

1.12×10

-3

7.89×10

-2

1.12×10

-1

eff

∆

std

Bor

opt

(a) No smearing

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

5.22×10

-5

2.41×10

-2

2.27×10

-1

4.59×10

-6

2.96×10

-2

2.78×10

-1

1.42×10

-5

1.67×10

-2

3.04×10

-2

eff

∆

std

Bor

opt

(b) Three smearing iterations

Figure 8.6.14:

Spectrum of

eff

with Wilson kernel for the standard (std), Boriçi

(Bor) and optimal (opt) construction at

4 on a smeared conﬁg-

uration with α = 0.5.

a Wilson and a staggered Wilson kernel. At the same time, in the case of Boriçi’s

and the optimal construction we often ﬁnd diﬀerences between the two kernel

operators of one to two orders of magnitude.

In practical applications the spectral bounds λ

min

and λ

max

of optimal domain

wall fermions are set on a per ensemble basis. As especially for small

and large en-

sembles one frequently observes values of

min

close to zero, one typically projects

out the lowest lying eigenvalues and treats them exactly in order to arrive at a

kernel operator with a narrower spectrum. As in our study we want to compare

the “baseline performance” of the various domain wall fermion formulations, we

decided to not add such an additional algorithmic component mixing into the

results of our benchmarks. However, as we also investigate large ensembles at

small

, the observed ratios of

min

/λ

min

can be very small in these cases, ren-

dering the optimal

∞

-approximation of the

sign

-function inadequate and of no

practical use. To mitigate this phenomenon, we decided to construct the

sign

function approximation on a per-conﬁguration rather than a per-ensemble basis.

This means that a small eigenvalue on a given conﬁguration does not render the

sign

-function approximation inadequate for the whole ensemble, but only on that

113

8 Staggered domain wall fermions

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

3.71×10

-3

1.51×10

-1

4.67×10

-1

7.69×10

-4

8.73×10

-2

8.59×10

-2

2.90×10

-4

1.17×10

-2

1.39×10

-2

eff

∆

std

Bor

opt

(a) No smearing

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

5.19×10

-5

2.29×10

-2

1.97×10

-1

1.16×10

-6

1.12×10

-3

5.97×10

-3

1.47×10

-5

4.40×10

-4

6.06×10

-4

eff

∆

std

Bor

opt

(b) Three smearing iterations

Figure 8.6.15:

Spectrum of

eff

with staggered Wilson kernel for the standard (std),

Boriçi (Bor) and optimal (opt) construction at

4 on a smeared

conﬁguration with α = 0.5.

particular conﬁguration. On the other hand this implies that in our benchmarks

the performance of optimal domain wall fermions tend to be overestimated and

can only be considered indicative of the performance of Chiu’s formulation of

domain wall fermions.

In general, we ﬁnd that the standard construction is outperformed by Boriçi’s

construction with respect to our chirality measures, while optimal domain wall

fermions show the best overall performance. A notable exception is the eﬀective

mass

eff

, where the optimal construction with a Wilson kernel is not so clearly

outperforming the competing variants. This is not surprising, considering that the

optimal rational function approximation of the

sign

-function deviates the most

min

(and

max

) and, thus, low-lying eigenvalues are typically not mapped very

accurately. However, as the size of the error decreases rapidly with increasing

values of N

, this problem is only present for small values of N

In this context the distribution of the lowest eigenvalue

min

= min

λ∈spec H

(8.6.5)

of the respective kernel operators

H = H

and

H = H

is of interest. In Fig. 8.6.20,

114

8.6 Numerical results

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

7.37×10

-2

1.82×10

-1

5.25×10

-1

8.26×10

-2

1.46×10

-1

1.46×10

-1

8.27×10

-2

1.74×10

-2

1.99×10

-2

eff

∆

std

Bor

opt

(a) Conﬁguration with Q = 0

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

3.71×10

-3

1.51×10

-1

4.67×10

-1

7.69×10

-4

8.73×10

-2

8.59×10

-2

2.90×10

-4

1.17×10

-2

1.39×10

-2

eff

∆

std

Bor

opt

(b) Conﬁguration with Q = 1

0.5

1.5

Re(λ)

-1

-0.5

0.5

Im(λ)

2.79×10

-3

1.08×10

-1

4.19×10

-1

4.41×10

-4

3.17×10

-2

3.17×10

-2

6.54×10

-5

7.37×10

-3

7.90×10

-3

eff

∆

std

Bor

opt

Figure 8.6.16:

Spectrum of

eff

with staggered Wilson kernel for the standard (std),

Boriçi (Bor) and optimal (opt) construction at

4 on conﬁgura-

tions with various Q .

-2 0 2 4

Re(λ)

-4

-2

Im(λ)

(a) Wilson kernel

-2 0 2 4

Re(λ)

-4

-2

Im(λ)

(b) Staggered Wilson kernel

Figure 8.6.17:

Spectrum of

in Boriçi’s construction at

4 on a smooth

topological conﬁguration with Q = 3.

we show the numerically determined cumulative distribution functions (CDFs)

min

for both the Wilson and staggered Wilson kernel at diﬀerent values of

When increasing

, we observe how the distribution moves toward larger values of

min

and pathologically small values become extremely unlikely. This is important,

as rational approximations to the

sign

-function become inaccurate for small argu-

ments. We also note that the corresponding CDFs for the Wilson and staggered

Wilson kernel are very similar.

115

8 Staggered domain wall fermions

−1

−0.5

0.5

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1

λ(m)

Fermion mass m

(a) Unsmeared conﬁguration

−1

−0.5

0.5

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1

λ(m)

Fermion mass m

(b) Smeared conﬁguration

−1

−0.5

0.5

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1

λ(m)

Fermion mass m

Figure 8.6.18: Spectral ﬂow of the Wilson kernel for conﬁgurations with Q = 2.

−1

−0.5

0.5

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1

λ(m)

Fermion mass m

(a) Unsmeared conﬁguration

−1

−0.5

0.5

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1

λ(m)

Fermion mass m

(b) Smeared conﬁguration

−1

−0.5

0.5

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1

λ(m)

Fermion mass m

Figure 8.6.19:

Spectral ﬂow of the staggered Wilson kernel for conﬁgurations with

Q = 2.

Finally, in Figs. 8.6.21 and 8.6.22 we show two examples of the

-dependence of

our chiral symmetry measures

∆

and

∆

. In the plot we show the median value

and the width of the distribution. Fixing the value

q =



1 − erf



√



, (8.6.6)

where

erf

(

)

refers to the error function, then we deﬁne the width using the

quantile and the

(

1 − q

)

-quantile. With our choice of

, we ﬁnd 68

3 % of the values

within the range. Our ﬁgures illustrate the superior chiral properties of the eﬀective

operator with a staggered Wilson kernel at suﬃciently large values of β. With the

use of smearing, the chiral properties are even further improved.

116

8.6 Numerical results

Kernel Construction N

eff

∆

2 5.47 × 10

−3

1.53 × 10

−1

6.72 × 10

−1

Standard 4 5.90 × 10

−4

6.87 × 10

−2

3.10 × 10

−1

6 1. 00 × 10

−4

2.30 × 10

−2

1.02 × 10

−1

2 5.73 × 10

−3

3.71 × 10

−1

1.16 × 10

Wilson Boriçi 4 8.56 × 10

−5

6.64 × 10

−2

3.00 × 10

−1

6 5.42 × 10

−6

1.57 × 10

−2

7.60 × 10

−2

2 8.30 × 10

−3

7.51 × 10

−1

1.16 × 10

Optimal 4 2.11 × 10

−3

3.59 × 10

−2

4.78 × 10

−2

6 8.71 × 10

−6

1.21 × 10

−3

1.74 × 10

−3

2 6.22 × 10

−3

1.48 × 10

−1

6.49 × 10

−1

Standard 4 7.18 × 10

−4

5.88 × 10

−2

2.88 × 10

−1

6 1.34 × 10

−4

2.01 × 10

−2

9.72 × 10

−2

2 6.38 × 10

−3

2.02 × 10

−1

2.92 × 10

−1

Staggered Wilson Boriçi 4 5.13 × 10

−5

5.45 × 10

−3

8.68 × 10

−3

6 5.91 × 10

−7

1.53 × 10

−4

2.69 × 10

−4

2 1.72 × 10

−2

2.32 × 10

−1

2.66 × 10

−1

Optimal 4 3.35 × 10

−5

2.23 × 10

−3

2.63 × 10

−3

6 5.02 × 10

−8

2.01 × 10

−5

2.36 × 10

−5

Table 8.6.1:

Median values for the various measures of chirality on unsmeared 32

conﬁgurations generated at

β =

8. For

eff

only conﬁgurations with

Q , 0 are considered.

117

8 Staggered domain wall fermions

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

CDF(λ

min

)

min

β = 3.2

β = 5.0

β = 12.8

(a) Wilson kernel

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

CDF(λ

min

)

min

β = 3.2

β = 5.0

β = 12.8

(b) Staggered Wilson kernel

Figure 8.6.20:

Cumulative distribution function

CDF

min

for the Hermitian ker-

nel operators.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

∆

1 / β

Wilson

Staggered Wilson

(a) Without smearing

0.5

1.5

2.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4

∆

1 / β

Wilson

Staggered Wilson

(b) With smearing

Figure 8.6.21:

Violation of normality

∆

as a function of 1

/β

for

eff

in the optimal

construction at N

= 2 (this ﬁgure is not part of Ref. [2]).

0.5

1.5

2.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4

∆

1 / β

Wilson

Staggered Wilson

(a) Without smearing

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1 1.2 1.4

∆

1 / β

Wilson

Staggered Wilson

(b) With smearing

Figure 8.6.22:

Violation of the Ginsparg-Wilson relation

∆

as a function of 1

/β

for %D

eff

in Boriçi’s construction at N

= 4.

118

8.7 Quenched quantum chromodynamics

0.5

1.5

2.5

Re(λ)

-1.5

-1

-0.5

0.5

1.5

Im(λ)

4.23×10

-1

1.42×10

2.38×10

3.99×10

2.57×10

2.60×10

∆

std

Bor

opt

(a) Wilson kernel

0.5

1.5

2.5

Re(λ)

-1.5

-1

-0.5

0.5

1.5

Im(λ)

8.07×10

-1

2.12×10

1.29×10

1.43×10

2.60×10

-1

4.07×10

-1

∆

std

Bor

opt

(b) Staggered Wilson kernel

Figure 8.7.1:

Spectrum of

eff

4 on a smeared 6

conﬁguration in QCD

at β = 6.

8.7 Quenched quantum chromodynamics

Although our study focuses on the setting of the Schwinger model, we also in-

vestigated the case of four-dimensional quantum chromodynamics. Our results

presented here are exploratory in nature, go beyond our discussion in Ref. [

] and

were ﬁrst presented in Ref. [1].

In Figs. 8.7.1 and 8.7.2 we can ﬁnd the eigenvalue spectrum of the eﬀective Dirac

operator in the standard, Boriçi’s and the optimal construction at

∈

{

4, 8

}

on a smeared four-dimensional 6

lattice. The

(

)

gauge conﬁguration under

consideration was generated at

β =

6 and then smeared with one step of the

 smearing prescription using a smearing parameter of

α =

65. As it is a

topologically trivial conﬁguration, the eﬀective mass

eff

is omitted in the ﬁgure

labels. Taking smearing into account, for the domain wall height we use M

= 1.4

in the case of the Wilson kernel and M

= 1 for the staggered Wilson kernel.

While in the case of the standard construction with a Wilson kernel we actually

observe smaller chiral symmetry violations compared to the staggered Wilson

kernel, for Boriçi’s and the optimal construction the use of the staggered Wilson

kernel results in clearly superior chiral properties. We often observe a reduction in

119

8 Staggered domain wall fermions

0.5

1.5

2.5

Re(λ)

-1.5

-1

-0.5

0.5

1.5

Im(λ)

8.33×10

-2

2.73×10

-1

5.39×10

-1

1.10×10

2.78×10

-2

3.15×10

-2

∆

std

Bor

opt

(a) Wilson kernel

0.5

1.5

2.5

Re(λ)

-1.5

-1

-0.5

0.5

1.5

Im(λ)

3.41×10

-1

7.71×10

-1

8.66×10

-2

8.94×10

-2

6.37×10

-4

9.55×10

-4

∆

std

Bor

opt

(b) Staggered Wilson kernel

Figure 8.7.2:

Spectrum of

eff

8 on a smeared 6

conﬁguration in QCD

at β = 6.

∆

and ∆

by more than an order of magnitude.

Although without considering more samples it is too early to judge the potential

beneﬁts of using staggered domain wall fermions in the setting of four-dimensional

quantum chromodynamics, our ﬁrst results are very encouraging.

8.8 Optimal weights

Before ending this chapter with our conclusions, we quote some example weight

factors

for optimal domain wall fermions as deﬁned in Sec. 8.3.3. We consider

the free-ﬁeld case in two dimensions and set

1 and

8. In the case of the

Wilson kernel we then ﬁx

min

1 and

max

3, while for the staggered Wilson

kernel we let

min

1 and

max

√

. The resulting weights

can be found in

Table 8.8.1.

120

8.9 Conclusions

s ω

(

min

= 1, λ

max

= 3

)



min

= 1, λ

max

√



1 0.989011284192743 0.996659816028010

2 0.908522120246430 0.971110743060917

3 0.779520722603956 0.925723982088869

4 0.641124364053574 0.869740043520870

5 0.519919928211430 0.813009342796322

6 0.427613177773962 0.763841917102527

7 0.366896221792507 0.728142270322088

8 0.337036936444470 0.709476563432225

Table 8.8.1: Weight factors ω

for the optimal domain wall fermion construction.

8.9 Conclusions

In this chapter, we explicitly constructed various formulations of staggered domain

wall fermions. We investigated their chiral properties in the free-ﬁeld case, on

quenched thermalized background ﬁelds in the Schwinger model and on smooth

topological conﬁgurations. With respect to our study, staggered domain wall fer-

mions work as advertised and generalize the domain wall fermion construction to

staggered kernels.

We also went beyond Adams’ original proposal of staggered domain wall fermi-

ons and generalized existing variants to the staggered case, such as Boriçi’s and

Chiu’s construction. This resulted in the formulation of truncated staggered do-

main wall fermions and optimal staggered domain wall fermions, which were pre-

viously not considered in the literature. Especially these new variants of staggered

domain wall fermions show signiﬁcantly improved chiral properties compared to

the traditional Wilson-based constructions, at least in the context of the Schwinger

model. In a setting where chiral symmetry plays an important role, the use of these

novel lattice fermion formulations can be potentially very advantageous.

While our results in the setting of a U

(

)

gauge theory are very encouraging,

the case of a

(

)

gauge group is not fully understood yet. As our ﬁrst results in

four-dimensional quantum chromodynamics are promising, they warrant further

investigations.

121



C 

C

In this thesis we aimed for a better understanding of the theoretical foundations,

computational properties and the practical applications of staggered Wilson, stag-

gered domain wall and staggered overlap fermions.

9.1 Summary

To summarize some of the core parts of this thesis we recall that in Chapter 3,

we gave an extensive review of the symmetries of staggered fermions and the

construction of ﬂavored mass terms. We discussed both Adams’ proposal for a

two-ﬂavor and Hoelbling’s proposal for a one-ﬂavor staggered Wilson term. In Sec.

3.6, we built upon these ideas and generalized the staggered Wilson term to allow

for arbitrary mass splittings in arbitrary even dimensions.

In Chapter 4, we estimated the gains in computational eﬃciency when using

staggered Wilson fermions instead of usual Wilson fermions for inverting the Dirac

matrix on a source, where we found a speedup factor of 4

–

6 in our particular

benchmark setting. We also analyzed the memory bandwidth requirements of

staggered Wilson fermions, quantiﬁed by the arithmetic intensity, and found that

they are comparable to other common staggered formulations.

In Chapter 5, we demonstrated the feasibility of spectroscopy calculations with

staggered Wilson fermions by studying the case of pseudoscalar mesons. We

adapted the operators used for spectroscopy with usual staggered fermions and

did a numerical study of the pion spectrum when turning the two-ﬂavor staggered

Wilson term on.

In Chapter 6, we reviewed the overlap construction and discussed Adams’ gen-

eralization to the case of a staggered kernel. In this context we did an analytical

veriﬁcation of the correctness of the continuum limit of the index and axial anomaly

for the staggered overlap Dirac operator in Sec. 6.4.

In Chapter 7, we analyzed the eigenvalue spectrum of the staggered Wilson

Dirac operator and related it to the computational eﬃciency of staggered overlap

123

9 Conclusions

fermions. We proposed an explanation to resolve the apparent discrepancies in

the speedup factors for the staggered Wilson kernel and for the staggered overlap

formulation.

Finally, in Chapter 8 we studied Adams’ proposal of staggered domain wall fer-

mions. In the process we generalized some variants of domain wall fermions to a

staggered kernel, investigated spectral properties and quantiﬁed chiral symmetry

violations of various domain wall formulations. We found that in the setting of

the Schwinger model the use of a staggered Wilson kernel results in signiﬁcantly

improved chiral properties.

9.2 Outlook

While staggered Wilson fermions and their derived formulations have both ad-

vantages and disadvantages compared to more established lattice fermion for-

mulations, we believe that their use can be beneﬁcial in some settings. In four

dimensions, the two ﬂavors of Adams’ staggered Wilson fermions can be naturally

used as a discretization of the light up and down quarks. Moreover, compared to

Wilson fermions, staggered Wilson fermions are at no disadvantage with respect to

ﬂavor-singlet physics [

]. Possible applications include the high precision determi-

nation of the

mass and the calculation of bulk quantities in thermodynamics

[

]. Furthermore, one can test universality, i.e. the independence of physical

results of the choice of the lattice action, with the help of new lattice fermion

formulations.

While the staggered Wilson kernel deﬁnes an interesting lattice fermion dis-

cretization in its own right, its derived chiral fermion formulations bear the poten-

tial of reducing the computational costs of simulations with chiral lattice fermions

signiﬁcantly. While staggered overlap fermions obey an exact chiral symmetry,

staggered domain wall fermions implement an approximate chiral symmetry in

a controlled manner. While the achievable speedup factor of staggered overlap

fermions compared to Neuberger’s overlap fermions has yet to be determined in a

realistic setting (cf. Chapter 7), for staggered domain wall fermions an improve-

ment of chirality by often more than an order of magnitude could be already shown

in the Schwinger model.

Besides Adams’ formulation, one may also ask about the potential of Hoelbling’s

single ﬂavor staggered action (see Sec. 3.5). As already pointed out in Ref. [

], it is

124

9.2 Outlook

important to note that in this case the hypercubic rotational symmetry is broken,

although the double rotation symmetry is preserved. This results in the need for two

gluonic counterterms to restore rotational invariance in unquenched simulations

[

] as the terms

+ G

and

+ G

can all appear with diﬀerent

coeﬃcients, where

µν

denotes the clover version of the gauge ﬁeld-strength tensor

(see e.g. Ref. [16]).

While this ﬁne-tuning is in principle possible, the simultaneous ﬁne-tuning of

coeﬃcients requires signiﬁcant eﬀort and makes the use of this action unlikely in

the context of practical applications in the near future. Perturbative methods allow

studying this ﬁne-tuning on an analytical level, although we expect the necessary

calculations to quickly become tedious due to the nontrivial structure of terms

of the form

α β

+ G

γδ

with

pairwise diﬀerent. While these issues do

not render Hoelbling’s formulation invalid, one can get an idea of the challenges

ahead from the many-year eﬀorts needed in studying similar issues in the context

of minimally doubled fermions of the Karsten-Wilczek kind [

221

222

], both on an

analytical [223] and a numerical level [224].

For the future, we hope that the practical potential of these new novel fermion

formulations is explored further, also in the case of full dynamical simulations. As

the technical properties of staggered Wilson fermions mimic the ones of Wilson

fermions (with the substitution

→ D

and

→ 

), it is expected that the

commonly employed algorithms for Wilson fermions can be adapted to staggered

Wilson fermions.

In particular rational and polynomial hybrid Monte Carlo methods [

225

–

227

]

would then allow the generation of two-ﬂavor full QCD conﬁgurations and render

any rooting procedure—together with any potential related problems—unnecessary.

However, practical implementations of these algorithms together with in-depth

studies of their theoretical and computational properties are still outstanding and

left for future work.

125

B

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