Theoretical and Computational Aspects of New Lattice Fermion Formulations
Theoretical and Computational Aspects
of New Lattice Fermion Formulations
Ph.D. Defense
Christian Zielinski
Division of Mathematical Sciences
Nanyang Technological University
Singapore
12
th
January 2017
Supervisor: Assoc. Prof. Wang Li-Lian
Additional supervision by Dr. David H. Adams and PD Dr. Christian Hoelbling
1 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Resulting publications
C. Hoelbling and C. Zielinski, “Staggered domain wall fermions,” PoS
LATTICE2016 (2016) 254, arXiv:1609.05114 [hep-lat].
C. Hoelbling and C. Zielinski, “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].
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].
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].
Remark: In lattice field theory authors are traditionally ordered by name
2 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Resulting presentations
1 2016, Workshop of the Collaborative Research Center Transregio SFB/TR-55,
University of Wuppertal, Wuppertal, Germany
2 2016, Spring Meeting of the German Physical Society,
University of Hamburg, Hamburg, Germany
3 2015, Invited Talk at the Theoretical Particle Physics Seminar,
University of Wuppertal, Wuppertal, Germany
4 2015, 33
rd
International Symposium on Lattice Field Theory,
Kobe International Conference Center, Kobe, Japan
5 2015, Conference on 60 Years of Yang-Mills Gauge Field Theories,
Institute of Advanced Studies, Nanyang Technological University, Singapore
6 2015, 9
th
International Conference on Computational Physics,
National University of Singapore, Singapore
7 2014, 5
th
Singapore Mathematics Symposium,
National University of Singapore, Singapore
8 2014, 32
nd
International Symposium on Lattice Field Theory,
Columbia University, New York City, United States
3 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Today’s talk with selected results
1 Introduction
Background
2 Fermion discretizations
Traditional formulations
Staggered Wilson fermions
3 Computational efficiency
Methodology
Numerical results
4 Hadron spectroscopy
Pseudoscalar mesons
Numerical results
5 Staggered domain wall fermions
Theory
Numerical results
6 Conclusions
4 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Introduction
Outline
1 Introduction
Background
2 Fermion discretizations
Traditional formulations
Staggered Wilson fermions
3 Computational efficiency
Methodology
Numerical results
4 Hadron spectroscopy
Pseudoscalar mesons
Numerical results
5 Staggered domain wall fermions
Theory
Numerical results
6 Conclusions
5 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Introduction
Background
Fundamental forces of nature
•
How do we describe the four fundamental forces in modern physics?
•
Gauge theories as fundamental building blocks
•
Observables invariant under transformations of some fundamental fields
•
All known fundamental forces of nature are gauge theories
•
From the four fundamental forces, the setting of this thesis is given by
•
Quantum chromodynamics (QCD)
•
Fundamental theory of the strong interaction
•
Binds quarks to form nuclear particles, such as the proton and neutron
•
Quantum electrodynamics (QED)
•
Fundamental relativistic theory of electrodynamics
•
Describes the interaction of light and matter
6 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Introduction
Background
Making predictions
•
How do we make predictions from the theory?
•
Application of analytical methods is limited
•
Predictions can be extracted numerically by
1 Discretizing the theory on a space-time grid (“lattice”)
2 Simulating the lattice formulation with the Monte Carlo method
3 Statistically evaluate the measurements and derive confidence interval
8 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Introduction
Background
Numerical approach
•
Numerical simulations are crucial for our understanding of QCD
•
Only framework for calculations from first principles
•
Simulations are computationally extremely challenging
•
Require the most powerful supercomputers
•
Many open problems with respect to doing these simulations
•
Among them a very old and notorious problem:
Can we construct “better” numerical formulations?
9 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Introduction
Background
Discretizing quantum field theories
•
In the lattice approach to quantum field theories, one discretizes Feynman’s
path integral formulation [Feynman ’48]
•
Evaluate expectation values of the form
hOi =
1
Z
Z
DΨDΨDU O
Ψ,Ψ, U
e
−S
f
[
Ψ,Ψ,U
]
−S
g
[U]
, (1)
Z =
Z
DΨDΨDU e
−S
f
[
Ψ,Ψ,U
]
−S
g
[U]
(2)
[here Ψ, Ψ are (anti-)fermionic fields, U is a gauge field, S
f
and S
g
are the action
functionals of the fermionic and gauge part of the theory
and
R
D
is a functional integral
over the space of possible field configurations]
•
In the discretized theory the ∞-dimensional path integral becomes
Z
DΨDΨDU −→
Z
Q
x∈Λ
dΨ(x )dΨ (x)dU (x ) (3)
over a (finite) lattice Λ
10 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Fermion discretizations
Outline
1 Introduction
Background
2 Fermion discretizations
Traditional formulations
Staggered Wilson fermions
3 Computational efficiency
Methodology
Numerical results
4 Hadron spectroscopy
Pseudoscalar mesons
Numerical results
5 Staggered domain wall fermions
Theory
Numerical results
6 Conclusions
11 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Fermion discretizations
Traditional formulations
Lattice fermion formulations
•
Fermions are particles with half-integral spin, e.g. quarks and electrons
•
All discretizations of fermions (i.e. of S
f
) suffer from some limitations
•
Most commonly used are Wilson and staggered fermions
•
At the core of the problem lies the Nielsen-Ninomiya No-Go theorem
[N & N ’80]
•
Can rigorously show that chiral symmetry (a fundamental symmetry of
massless fermions) implies the presence of spurious states on the lattice,
so-called fermion doublers
12 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Fermion discretizations
Traditional formulations
Naïve and Wilson fermions
•
Naïve discretization of fermions reads
S
N
= a
4
X
n∈Λ
Ψ(n)(D
N
+ m) Ψ(n) (4)
with lattice spacing a, fermion field Ψ, Dirac operator D
N
= γ
µ
∇
µ
, Dirac
matrices γ
µ
∈ C
4×4
, difference operator ∇
µ
=
1
2a
(T
µ+
− T
µ−
) with
T
µ±
χ(x ) = U
±µ
(x) χ(x ± ˆµ) and mass m
•
However, naïve discretization contains 15 fermion doublers
•
For Wilson fermions [Wilson ’74], we introduce an additional O(a) term
D
N
−→ D
w
= γ
µ
∇
µ
+
ar
2
∆, ∆ = ∇
2
µ
, r ∈ (0,1] (5)
•
Describes one flavor (= species), i.e. no fermion doublers
•
Breaks chiral symmetry in the massless case, i.e. {γ
5
,D
w
} 6= 0
13 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Fermion discretizations
Traditional formulations
Staggered fermions
•
For staggered fermions, one does a “spin diagonalization” using a local
unitary transformation [Kogut & Susskind ’75]
•
Can reduce to an action for a single-component spinor
S
st
= a
4
X
n∈Λ
χ(n)(D
st
+ m) χ(n) (6)
with D
st
= η
µ
∇
µ
and η
µ
χ(n) = (−1)
n
1
+...+n
µ−1
χ(n)
•
Respects an exact (residual) chiral symmetry
•
Describes four flavors (= species), i.e. presence of three fermion doublers
•
Staggered fermions are computationally efficient as χ is a one-component
field, while the original Ψ is a four-component field
14 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Fermion discretizations
Staggered Wilson fermions
Staggered Wilson fermions
•
Can we further reduce the number of doublers and construct a novel
staggered fermion formulation?
•
Combine the approaches of Wilson and staggered fermions
•
Gives rise to staggered Wilson fermions [Adams ’10, ’11]
•
Introduction of a Wilson-like term for staggered fermions
•
Reduces the number of species from four to two
•
Breaks the residual chiral symmetry of staggered fermions
•
Results in technical properties comparable to Wilson fermions
15 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Fermion discretizations
Staggered Wilson fermions
The Dirac operator
•
Proposal:
S
sw
= a
4
X
n∈Λ
χ(n)(D
sw
+ m) χ(n) (7)
with D
sw
= D
st
+ W
st
and staggered Wilson term W
st
=
r
a
(
1
− Γ
55
Γ
5
)
[Adams ’10, ’11]
•
Operator definitions
•
Γ
55
χ(n) = (−1)
n
1
+...+n
4
χ(n) with Γ
55
' γ
5
⊗ γ
5
, γ
5
= γ
1
γ
2
γ
3
γ
4
•
Γ
5
= η
5
C with Γ
5
' γ
5
⊗
1
+ O
a
2
in spin ⊗ flavor
•
C = (C
1
C
2
C
3
C
4
)
sym
with C
µ
=
1
2
(T
µ+
+ T
µ−
)
•
η
5
= η
1
η
2
η
3
η
4
16 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Computational efficiency
Outline
1 Introduction
Background
2 Fermion discretizations
Traditional formulations
Staggered Wilson fermions
3 Computational efficiency
Methodology
Numerical results
4 Hadron spectroscopy
Pseudoscalar mesons
Numerical results
5 Staggered domain wall fermions
Theory
Numerical results
6 Conclusions
17 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Computational efficiency
Methodology
Motivation
•
One of the potential major advantages of staggered Wilson fermions over
Wilson fermions is an increased computational efficiency
•
We did a numerical study to investigate the computational efficiency
•
We implemented staggered Wilson fermions in the Chroma/QDP
software package [Edwards et al. ’05]
•
Setting is quenched lattice QCD (i.e. neglecting quark loops)
with 200 gauge configurations
•
Some results published in PoS LATTICE2013 (2014) 353
18 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Computational efficiency
Methodology
Measuring the computational efficiency
•
Computational cost of simulations dominated by inverting the Dirac operator
D on a source χ:
DΨ = χ (8)
•
Quantify efficiency by comparing the cost for inversion for staggered Wilson
with usual Wilson fermions
•
Do comparison at fixed physical parameter, i.e. pion mass m
π
(the pion is a composite particle made up of a quark and an antiquark)
•
Use conjugate gradient method for the normal equation D
†
DΨ = D
†
χ
•
Cost can be characterized by:
cost = (#iters) ×(cost per iteration)
| {z }
∝ cost mat-vec mult.
(9)
•
Take the ratio:
cost
w
cost
sw
=
(#iters)
w
(#iters)
sw
| {z }
depends on m
π
×
(cost mat-vec mult.)
w
(cost mat-vec mult.)
sw
| {z }
depends on fermion matrix
(10)
19 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Computational efficiency
Methodology
Matrix-vector multiplications
•
For small residual ε expected [Shewchuk ’94]
#iters ∝
√
κlog (2/ε) , κ = |λ
max
/λ
min
| (11)
•
Hence we expect as a rough estimate
(#iters)
w
(#iters)
sw
≈
q
κ
w
κ
sw
, (12)
but need to measure numerically
•
Costs for matrix-vector mult. can be estimated from FLOP counts:
(FLOPs)
w
(FLOPs)
sw
=
4 ×1392 FLOPs/site
1743 FLOPs/site
≈ 3.2, (13)
which we conservatively estimate to be O (2 − 3)
20 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Computational efficiency
Numerical results
Ratio of number of CG iterations
0
0.5
1
1.5
2
2.5
3
0.04 0.06 0.08 0.1 0.12 0.14 0.16
κ
1/2
ratio & CG iters ratio vs. m
π
2
(16
3
×32, β = 6)
κ
1/2
ratio
CG ratio, ε = 10
−06
CG ratio, ε = 10
−10
CG ratio, ε = 10
−14
Figure: Only mild dependence on pion mass
21 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Computational efficiency
Numerical results
Computational efficiency for 16
3
× 32, β = 6
We find
cost
W
cost
SW
=
(# CG iters)
W
(# CG iters)
SW
| {z }
≈ 2
×
(cost MV mult.)
W
(cost MV mult.)
SW
| {z }
≈ 2−3
≈ 4 − 6 (14)
Efficiency
Staggered Wilson is 4 − 6 times more efficient than usual Wilson for inverting the
Dirac operator in this setting
22 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Computational efficiency
Numerical results
Computational efficiency for 20
3
× 40, β = 6
0
0.5
1
1.5
2
2.5
3
0.04 0.06 0.08 0.1 0.12 0.14 0.16
κ
1/2
ratio & CG iters ratio vs. m
π
2
(20
3
×40, β = 6)
κ
1/2
ratio
CG ratio, ε = 10
−06
CG ratio, ε = 10
−10
CG ratio, ε = 10
−14
Figure: Larger physical volume ⇒ improves at small m
π
, otherwise unchanged
23 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Computational efficiency
Numerical results
Computational efficiency for 20
3
× 40, β = 6.136716
0
0.5
1
1.5
2
2.5
3
0.04 0.06 0.08 0.1 0.12 0.14 0.16
κ
1/2
ratio & CG iters ratio vs. m
π
2
(20
3
×40, β = 6.14)
κ
1/2
ratio
CG ratio, ε = 10
−06
CG ratio, ε = 10
−10
CG ratio, ε = 10
−14
Figure: Smaller lattice spacing ⇒ significant improvement in efficiency
24 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Computational efficiency
Numerical results
Conclusions
•
Two-flavor staggered Wilson fermions more efficient by a factor
of O (4 − 6) for inverting the Dirac operator at β = 6
•
Dependence on physical volume and lattice spacing has been investigated
•
Speed-up factor increases with decreasing lattice spacing
•
Mostly unchanged for increasing volume
•
Took a few months of computing time on two NTU clusters
25 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Hadron spectroscopy
Outline
1 Introduction
Background
2 Fermion discretizations
Traditional formulations
Staggered Wilson fermions
3 Computational efficiency
Methodology
Numerical results
4 Hadron spectroscopy
Pseudoscalar mesons
Numerical results
5 Staggered domain wall fermions
Theory
Numerical results
6 Conclusions
26 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Hadron spectroscopy
Pseudoscalar mesons
Hadron spectroscopy
•
One of the major applications of lattice QCD is hadron spectroscopy
•
Hadrons are composite particles made up of quarks, e.g. protons and pions
•
Hadron spectroscopy is the derivation of the masses of hadronic particles
•
Can we employ staggered Wilson fermions?
•
For usual staggered fermions, meson and baryon rest-frame operators
are known [Golterman & Smit, Sharpe, ...]
•
Can adapt operators to staggered Wilson fermions
•
Physical interpretation changes!
•
We illustrate spectrum calculations for the case of pseudoscalar mesons
•
Example: Pions (π), which are produced in the Earth’s atmosphere
due to high energy cosmic ray protons
27 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Hadron spectroscopy
Pseudoscalar mesons
Pseudoscalar mesons
•
The case of usual staggered fermions
•
There are 16 pseudoscalar mesons
•
They have spin⊗ flavor structure γ
5
⊗ ξ
F
with ξ
F
∈ {
1
, ξ
5
, ξ
µ
, ξ
µ
ξ
5
, ξ
µ
ξ
ν
}
•
The ξ
µ
are a representation of the Clifford algebra
in flavor space, i.e. {ξ
µ
,ξ
ν
} = 2δ
µν
1
•
We can find two types of states [Bae et al. ’08]
•
Some states propagate with a factor of (−1)
t
and some do not
•
In general the timeslice operators excite two states
•
If the operator couples to γ
S
⊗ ξ
F
, then also to the
“time-parity partners” γ
4
γ
5
γ
S
⊗ ξ
4
ξ
5
ξ
F
(here γ
S
= γ
5
or γ
S
= γ
4
γ
5
)
28 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Hadron spectroscopy
Pseudoscalar mesons
Staggered Wilson Dirac operator
•
The time-time correlation function can be parametrized by
R
+
cosh[m
+
(N
t
/2 − t)] + (−1)
t
R
−
cosh[m
−
(N
t
/2 − t)] (15)
if one-particle states dominate [Goltermann ’86]
•
As the staggered Wilson term makes two flavors heavy,
ξ
F
is of the following structure
ξ
F
∼
=
light+light light+heavy
heavy+light heavy+heavy
(16)
•
In the continuum limit heavy contributions decouple
•
The “light+light” part of ξ
F
then determines the physical interpretation
29 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Hadron spectroscopy
Pseudoscalar mesons
Continuum pseudoscalar mesons
•
Out of the 16 pseudoscalar mesons, 8 become heavy
•
In the continuum limit we are left with eight pseudoscalar mesons
•
We find four physical particles with a two-fold degeneracy
Flavor structure ξ
F
Particle composition Particle interpretation
ξ
k
ξ
4
qσ
k
q π triplet
ξ
i
ξ
j
ξ
5
qq η singlet
1
Table: Mesons formed by the two physical flavors q ≡ (q
1
,q
2
)
|
30 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Hadron spectroscopy
Numerical results
Numerical tests
•
Quenched study on a 16
3
× 64 lattice at β = 6
•
Cross-checked our implementation for usual staggered fermions (r = 0)
•
Match the masses reported in Bae et al. ’08
•
Numerical results with staggered Wilson fermions (r > 0)
•
We find m
2
∝ m
q
in accordance with chiral perturbation theory
•
All degeneracies as expected
31 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Hadron spectroscopy
Numerical results
Usual staggered fermions: Pseudoscalar meson spectrum
0
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
q
Our γ
5
x ξ
5
Our γ
5
x ξ
i5
Our γ
5
x ξ
i4
Our γ
5
x ξ
4
Lit. γ
5
x ξ
5
Lit. γ
5
x ξ
i5
Lit. γ
5
x ξ
i4
Lit. γ
5
x ξ
4
0
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
q
Our γ
45
x 1
Our γ
45
x ξ
45
Our γ
45
x ξ
ij
Our γ
45
x ξ
i
Lit. γ
45
x 1
Lit. γ
45
x ξ
45
Lit. γ
45
x ξ
ij
Lit. γ
45
x ξ
i
Figure: Pseudoscalar meson mass m
2
(m
q
)
32 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Hadron spectroscopy
Numerical results
Staggered Wilson fermions: Light spectrum
0
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
2
Quark bare mass m
q
γ
5
x ξ
5
γ
45
x 1
γ
5
x ξ
i4
γ
45
x ξ
ij
33 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Hadron spectroscopy
Numerical results
Conclusions
•
We showed how the pseudoscalar meson spectrum changes
after the introduction of the staggered Wilson term
•
We determined the physical part of the spectrum and identified
the corresponding particles
•
We demonstrated the feasibility of numerical spectrum calculations
with staggered Wilson fermions
34 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Staggered domain wall fermions
Outline
1 Introduction
Background
2 Fermion discretizations
Traditional formulations
Staggered Wilson fermions
3 Computational efficiency
Methodology
Numerical results
4 Hadron spectroscopy
Pseudoscalar mesons
Numerical results
5 Staggered domain wall fermions
Theory
Numerical results
6 Conclusions
35 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Staggered domain wall fermions
Theory
Domain wall fermions with a staggered kernel
•
Possible to use the overlap operator if chiral symmetry is physically relevant
•
Continuum chiral symmetry {γ
5
,D} = 0 replaced by {γ
5
,D} = aDγ
5
D
•
Expensive to use overlap formulation
•
Requires O (10 − 100) times more resources than Wilson fermions
•
Domain wall fermions as an alternative?
•
Formulate fermions in d dimensions by means of a (d + 1)-dim. theory
•
Can we improve the properties with a staggered kernel?
•
Usual staggered fermions not applicable to domain wall construction
•
Proposal of staggered domain wall fermions [Adams ’10, ’11]
•
What are the properties of these novel staggered domain wall fermions?
•
Schwinger model results published in Phys. Rev. D94 (2016) no.1, 014501
•
Further insights and QCD results published in
PoS LATTICE2016 (2016) 254
36 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Staggered domain wall fermions
Theory
Staggered Wilson kernel in two dimension
•
Our setting is two-dimensional QED, i.e. the Schwinger model [Schwinger ’62]
•
Numerically simple enough for complete eigenvalue spectra
•
The staggered Wilson Dirac operator reads
D
sw
(m
f
) = D
st
+ m
f
+ W
st
(17)
•
In two dimensions we consider the staggered Wilson term
W
st
=
r
a
(
1
+ i η
3
C) , r > 0 (18)
[with (x ) = (−1)
x
1
+x
2
, η
3
= η
1
η
2
, C =
1
2
{C
1
,C
2
}, C
µ
=
1
2
T
µ
+ T
†
µ
]
37 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Staggered domain wall fermions
Theory
Domain wall fermions (DWFs) in two dimensions
•
Domain wall operator in 3d reads [Kaplan ’92, Shamir ’93, Furman & Shamir ’94]
ΨD
dw
Ψ =
X
N
s
s=1
Ψ
s
D
+
w
Ψ
s
− P
−
Ψ
s+1
− P
+
Ψ
s−1
(19)
with P
±
=
1
2
(
1
± γ
3
)
, D
±
w
= a
3
D
w
(−M
0
) ±
1
and domain wall height M
0
= 1
•
Boriçi’s modification follows from replacement [Boriçi ’99]
P
±
Ψ
s∓1
−→ −D
−
w
P
±
Ψ
s∓1
(20)
•
Chiu introduces weights {ω
s
} for optimal DWFs [Chiu ’02]
D
±
w
−→ D
±
w
(s) = a
3
ω
s
D
w
(−M
0
) ±
1
(21)
•
Staggered versions are obtained by replacement rule [Adams ’11]
D
w
−→ D
sw
, γ
3
−→ (x)
∼
=
γ
3
⊗ γ
3
(22)
38 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Staggered domain wall fermions
Theory
Effective low energy Dirac operator
•
Combine left- and right-handed modes into a 2d spinor
q = P
+
Ψ
N
s
+ P
−
Ψ
1
, q = Ψ
1
P
+
+ Ψ
N
s
P
−
(23)
•
Define two-dimensional low energy effective action
S
eff
=
X
x
q (x )D
eff
q (x ), D
−1
eff
(x, y) = hq (x) q (y )i (24)
•
Explicitly [Kikukawa & Noguchi ’99]
D
eff
=
1 + m
2
1
+
1 − m
2
γ
3
T
N
s
+
− T
N
s
−
T
N
s
+
+ T
N
s
−
(25)
with T
±
=
1
± a
3
H and
H =
γ
3
D
w
2 ·
1
+ a
3
D
w
(standard), H = γ
3
D
w
(Bori¸ci) (26)
39 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Staggered domain wall fermions
Numerical results
Numerical setting
•
Computations are done on a 20
2
× N
s
lattice at β = 5
•
Low-lying free field spectrum approximately invariant by considering
%
D
eff
with
% =
(
2M
0
− a
3
M
2
0
for standard construction,
2M
0
for Bori¸ci
0
s/Chiu
0
s construction.
(27)
•
We quantify chiral symmetry violations via [Dürr et al. ’05, Dürr et al. ’11]
•
Approximate zero modes give rise to m
eff
= min
λ∈spec H
|λ|
•
Use on topological nontrivial configurations [Gadiyak et al. ’00]
•
Deviation from normality ∆
N
=
D, D
†
∞
•
Normality necessary for chirality [Kerler ’00]
•
Violation of the Ginsparg-Wilson relation ∆
GW
=
{γ
3
,D} − 2%
−1
Dγ
3
D
∞
•
Note that lim
N
s
→∞
D
eff
= D
ov
, with overlap operator D
ov
[Neuberger ’98]
40 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Staggered domain wall fermions
Numerical results
Effective operator at N
s
= 2
0
0.5
1
1.5
2
Re(λ)
-1
-0.5
0
0.5
1
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
0
1.12×10
-1
1.43×10
0
1.80×10
0
m
eff
∆
N
∆
GW
std
Bor
opt
(a) Wilson kernel
0
0.5
1
1.5
2
Re(λ)
-1
-0.5
0
0.5
1
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
m
eff
∆
N
∆
GW
std
Bor
opt
(b) Staggered Wilson kernel
Figure: Spectrum of %D
eff
41 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Staggered domain wall fermions
Numerical results
Approaching the continuum
0
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1 1.2 1.4
∆
GW
1 / β
Wilson
Staggered Wilson
(a) Without smearing
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
∆
GW
1 / β
Wilson
Staggered Wilson
(b) With smearing
Figure: Violation of the GW relation ∆
GW
for %D
eff
in Boriçi’s construction at N
s
= 4
42 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Staggered domain wall fermions
Numerical results
Results and summary
•
We gave an explicit construction of staggered domain wall fermions
•
We generalized Boriçi’s and the optimal construction to the staggered case
•
In the Schwinger model they appear to work as advertised
•
Chiral properties on par with or even better than Wilson based formulations
•
Application to four-dimensional quantum chromodynamics
•
First results are encouraging and warrant further investigations
43 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Conclusions
Outline
1 Introduction
Background
2 Fermion discretizations
Traditional formulations
Staggered Wilson fermions
3 Computational efficiency
Methodology
Numerical results
4 Hadron spectroscopy
Pseudoscalar mesons
Numerical results
5 Staggered domain wall fermions
Theory
Numerical results
6 Conclusions
44 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Conclusions
Summary of original contributions (discussed today)
1 Generalization of staggered Wilson terms to arbitrary mass splittings in
arbitrary even dimensions
2 Quantitative analysis of the computational efficiency of staggered Wilson
fermions and their memory bandwidth requirements
3 Adaption of hadron spectroscopy methods to staggered Wilson fermions and
the determination of the pseudoscalar meson spectrum
4 Analysis of the continuum limit of the index and axial anomaly for the
staggered overlap Dirac operator
5
Investigation of the Wilson and staggered Wilson eigenvalue spectrum and its
connection to the computational efficiency of the overlap construction
6
Study of chiral symmetry violations of known and newly proposed variants of
usual and staggered domain wall fermions
45 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Conclusions
Summary of original contributions (discussed in the thesis)
1 Generalization of staggered Wilson terms to arbitrary mass splittings in
arbitrary even dimensions
2 Quantitative analysis of the computational efficiency of staggered Wilson
fermions and their memory bandwidth requirements
3 Adaption of hadron spectroscopy methods to staggered Wilson fermions and
the determination of the pseudoscalar meson spectrum
4 Analysis of the continuum limit of the index and axial anomaly for the
staggered overlap Dirac operator
5
Investigation of the Wilson and staggered Wilson eigenvalue spectrum and its
connection to the computational efficiency of the overlap construction
6
Study of chiral symmetry violations of known and newly proposed variants of
usual and staggered domain wall fermions
46 / 47
Theoretical and Computational Aspects of New Lattice Fermion Formulations
Conclusions
Conclusions
Staggered Wilson, staggered domain wall and staggered overlap fermions
•
Define theoretically sound discretizations of fermions on the lattice
•
Have practical advantageous in certain settings
Advantages
•
Computationally more efficient than Wilson fermions in certain settings
•
Reduced number of fermion doublers compared to staggered fermions
•
Allow formulation of staggered chiral fermion formulations
•
Improved chiral properties of staggered domain wall fermions
•
Staggered overlap operator suitable to study e.g. topological aspects
(See analytical results in PoS LATTICE2013 (2014) 462)
Disadvantages
•
Complex construction, nontrivial framework for physical simulations
•
Highly efficient implementation on massively parallel systems difficult
47 / 47
