Staggered domain wall fermions
Christian Hoelbling
∗
Department of Physics, University of Wuppertal, D-42119 Wuppertal, Germany
E-mail: hch@uni-wuppertal.de
Christian Zielinski
Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371 &
Department of Physics, University of Wuppertal, D-42119 Wuppertal, Germany
E-mail: zielinski@pmail.ntu.edu.sg
We construct domain wall fermions with a staggered kernel and investigate their spectral and
chiral properties numerically in the Schwinger model. In some relevant cases we see an improve-
ment of chirality by more than an order of magnitude as compared to usual domain wall fermions.
Moreover, we present first results for four-dimensional quantum chromodynamics, where we also
observe significant reductions of chiral symmetry violations for staggered domain wall fermions.
34th annual International Symposium on Lattice Field Theory
24-30 July 2016
University of Southampton, UK
∗
Speaker.
c
Copyright owned by the author(s) under the terms of the Creative Commons
Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/
arXiv:1609.05114v2 [hep-lat] 20 Dec 2016
Staggered domain wall fermions Christian Hoelbling
1. Introduction
When implementing chiral fermions on the lattice, domain wall fermions [
1
,
2
,
3
] are a well-
known alternative to the computationally very expensive overlap construction [
4
,
5
,
6
]. By means
of massive interacting fermions in
d + 1
dimensions, one can formulate lattice fermions with an
approximate chiral symmetry in
d
dimensions. The accuracy of this approximate symmetry is
controlled by the extent of the extra dimension, where the limit of an infinite extent can be concisely
described by an overlap operator. Domain wall fermions are computationally cheaper, permitting a
simpler approach to parallelization and allowing for an easier tunneling between different topological
sectors as compared to overlap fermions. This, however, comes at the price of replacing the exact
chiral symmetry of overlap fermions with an approximate one.
Traditionally, domain wall fermions are used with a Wilson kernel as the naïve application of
the formalism to the staggered kernel fails due to the lack of some technical properties within the
staggered framework. This problem was eventually overcome with Adams’ proposal of staggered
domain wall fermions [
7
,
8
], where one introduces a modified kernel operator, i.e. so-called staggered
Wilson fermions. The resulting novel kernel operator is constructed by adding a suitable flavored
mass term [
9
] to staggered fermions (see also Ref. [
10
]). As staggered Wilson fermions appear to
be computationally more efficient than the Wilson kernel [11], one can hope for a computationally
cheaper and possibly more chiral formulation. Besides staggered domain wall fermions, staggered
Wilson fermions also permit the formulation of staggered overlap fermions [
12
,
13
] with a well-
defined index [14].
In this report, we present selected results of our investigations of the spectral properties and
chiral symmetry violations of domain wall fermions with Wilson and staggered Wilson kernel
operators in the Schwinger model [
15
] as recently discussed in Ref. [
16
]. In addition, we present
first results for the case of four-dimensional quenched quantum chromodynamics.
2. Formulation
We begin with a quick review of the
d
-dimensional kernel operators and the
(d + 1)
-dimensional
domain wall fermion construction (
d = 2,4
). We denote the lattice spacing by
a
in the first
d
dimensions and by
a
d+1
in the extra dimension. The
γ
µ
matrices (
µ = 1, ...,d
) refer to a
representation of the Dirac algebra, where we use the notation γ
d+1
for the chirality matrix.
Kernel operators.
We denote the
d
-dimensional Wilson Dirac operator with bare fermion mass
m
f
by D
w
(m
f
). For the definition of staggered Wilson fermions, we use our notation introduced in
Ref. [
16
] and write the Dirac operator as
D
sw
(m
f
) = D
st
+ m
f
+W
st
. Here
D
st
= η
µ
∇
µ
is the usual
staggered Dirac operator and our choice of the staggered Wilson term can be compactly written as
W
st
=
r
a
1
+ λW
1···d
st
, λ = (−1)
d+2
2
. (2.1)
In four dimensions our
W
st
equals Adams’ staggered Wilson term, for the two-dimensional case we
refer to our discussion in Ref. [
16
]. In the following discussion, we set the
d
-dimensional lattice
spacing to a = 1 and the (staggered) Wilson parameter to r = 1.
1
Staggered domain wall fermions Christian Hoelbling
0
0.5
1
1.5
2
Re(λ)
-1
-0.5
0
0.5
1
Im(λ)
9.48×10
-5
1.47×10
-2
5.10×10
-2
2.12×10
-6
5.59×10
-3
2.00×10
-2
2.03×10
-7
9.01×10
-5
1.15×10
-4
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)
1.51×10
-4
1.72×10
-2
6.42×10
-2
4.70×10
-7
1.23×10
-4
1.20×10
-4
3.43×10
-9
1.38×10
-6
1.48×10
-6
m
eff
∆
N
∆
GW
std
Bor
opt
(b) Staggered Wilson kernel
Figure 1:
Spectrum of
ρD
eff
for the standard (std), Boriçi (Bor) and optimal (opt) construction at
N
s
= 8
on
a 20
4
configuration with topological charge Q = 3 in the Schwinger model at β = 5.
Bulk operators.
The bulk Dirac operator in the standard (i.e. original) formulation of domain
wall fermions reads
ΨD
dw
Ψ =
N
s
∑
s=1
Ψ
s
D
+
w
Ψ
s
−P
−
Ψ
s+1
−P
+
Ψ
s−1
, (2.2)
where the extent of the extra dimension is denoted by
N
s
, the
(d + 1)
-dimensional fermion fields
Ψ
and
Ψ
have an index
s
for the additional spatial coordinate, we introduced
D
±
w
= a
d+1
D
w
(−M
0
)±
1
,
the domain wall height is given by
M
0
and the chiral projectors are defined as
P
±
= (
1
±γ
d+1
)/2
.
In the extra dimension we impose the boundary conditions
P
+
(Ψ
0
+ mΨ
N
s
) = 0, P
−
(Ψ
N
s
+1
+ mΨ
1
) = 0, (2.3)
where the fermion mass is controlled by the parameter
m
. Finally, by letting
q = P
+
Ψ
N
s
+ P
−
Ψ
1
and q = Ψ
1
P
+
+ Ψ
N
s
P
−
, we can define d-dimensional fermion fields from the boundary.
Besides this original formulation, we also consider two common
O(a
d+1
)
modifications.
Boriçi’s construction [17] follows from the replacement
P
+
Ψ
s−1
→ −D
−
w
P
+
Ψ
s−1
, P
−
Ψ
s+1
→ −D
−
w
P
−
Ψ
s+1
, (2.4)
while Chiu introduces additional weight factors for the formulation of optimal domain wall fermions
[18] in order to improve chiral properties.
Effective operators.
Following the derivation given in Refs. [
19
,
17
] and integrating out
N
s
−1
heavy modes, we can define a
d
-dimensional low-energy effective action
S
eff
=
∑
x
q(x)D
eff
q(x)
,
2
Staggered domain wall fermions Christian Hoelbling
where
D
−1
eff
(x,y) =
h
q(x)q (y)
i
. Note that in order to cancel the diverging contributions from the
heavy fermions in the chiral limit
N
s
→ ∞
, one introduces suitable pseudofermion fields. The
effective operator can then be written in closed form as
D
eff
=
1 + m
2
1
+
1 −m
2
γ
d+1
T
N
s
+
−T
N
s
−
T
N
s
+
+ T
N
s
−
, (2.5)
where
T
±
=
1
±a
d+1
H
. In the case of the standard construction
H
equals the modified kernel
operator
H
m
= γ
d+1
D
m
(−M
0
)
with
D
m
= D
w
/(2 ·
1
+ a
d+1
D
w
)
. For Boriçi’s construction, we
find the standard kernel
H
w
= γ
d+1
D
w
(−M
0
)
and that the effective Dirac operator equals the
polar decomposition approximation of Neuberger’s overlap operator. Finally, for Chiu’s optimal
construction the fraction on the right hand side of Eq.
(2.5)
is replaced by Zolotarev’s optimal
rational function approximation [
20
] of
signH
w
. In all cases the corresponding overlap operator is
given by D
ov
= lim
N
s
→∞
D
eff
.
As discussed in full detail in Ref. [
16
], in order to ensure the same scale of all effective operators,
we let ρ = 2ω with
ω =
(
M
0
−
1
2
a
d+1
M
2
0
for the standard construction,
M
0
for Boriçi’s/Chiu’s construction
(2.6)
and consider ρD
eff
and ρD
ov
in all our numerical investigations.
Staggered versions.
As shown in Ref. [
8
], one can obtain a staggered version of domain wall
fermions by means of a replacement rule. In our setting, it takes the form
D
w
→D
sw
,
γ
d+1
→ε
with
ε (x) = (−1)
x
1
/a+···+x
d
/a
, allowing a full generalization of the previous discussion to the staggered
case.
3. Numerical results
For the numerical part of our work, we begin by introducing measures of chirality followed
by the discussion of selected results in the setting of the Schwinger model and four-dimensional
quantum chromodynamics. For the remainder of this report, we consider the massless case m = 0.
Measures of chirality.
In order to quantify chiral symmetry violations for the effective Dirac
operators, we define three different measures. The first is the effective mass
m
eff
as used in Ref. [
21
].
Using periodic boundary conditions and considering a topologically nontrivial configuration, we let
m
eff
= min
λ ∈spec H
|
λ
|
= min
Λ∈specD
†
D
√
Λ. (3.1)
As chiral properties imply the normality of the Dirac operator [
22
], we also consider deviations
from operator normality as given by
∆
N
=
D,D
†
∞
. Finally, we can directly measure violations
of the Ginsparg-Wilson relation by
∆
GW
=
{
γ
d+1
,D
}
−ω
−1
Dγ
d+1
D
∞
, (3.2)
where in the staggered case
γ
d+1
is replaced by
ε
. We note that for the overlap operator, all our
measures vanish in exact arithmetics.
3
Staggered domain wall fermions Christian Hoelbling
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
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8 1 1.2 1.4
∆
GW
1 / β
Wilson
Staggered Wilson
(b) With smearing
Figure 2: Measure ∆
GW
for ρD
eff
in the optimal construction at N
s
= 4 in the Schwinger model.
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
∆
N
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
0 0.2 0.4 0.6 0.8 1 1.2 1.4
∆
N
1 / β
Wilson
Staggered Wilson
(b) With smearing
Figure 3: Measure ∆
N
for ρD
eff
in Boriçi’s construction at N
s
= 4 in the Schwinger model.
Schwinger model. Let us now discuss a few selected results for the two-dimensional Schwinger
model. For our numerical study we use the canonical choice M
0
= 1.
In Fig. 1, we find the eigenvalue spectra of the effective operator
ρD
eff
for various constructions
on an exemplary gauge configuration. We observe that already for the small value of
N
s
= 8
the
spectrum very closely resembles that of the corresponding overlap operator. Comparing the effective
operators with the usual Wilson and the staggered Wilson operator, we note that in the case of
the standard construction
m
eff
,
∆
N
and
∆
GW
are typically of comparable magnitude. For Boriçi’s
and Chiu’s optimal construction, however, we find that the staggered construction shows notably
improved chiral properties compared to the Wilson case.
We are particularly interested in the chiral properties when approaching the continuum limit.
To this end we generated seven ensembles with the setup of Ref. [
23
] with
10
3
configurations each,
namely
8
2
at
β = 0.8
,
12
2
at
β = 1.8
,
16
2
at
β = 3.2
,
20
2
at
β = 5.0
,
24
2
at
β = 7.2
,
28
2
at
β = 9.8
and
32
2
at
β = 12.8
. The values for
β
were chosen so that the physical volume is kept fixed. We
consider both unsmeared and three-step APE smeared configurations [
24
] with a smearing parameter
of α = 0.5, resulting in 14000 configurations in total.
In Figs. 2 and 3, we find two examples for the behavior of
∆
GW
and
∆
N
when
β
is varied. In the
figures, we plot the median value and the
68.3%
-width of the distribution. We find that domain wall
fermions with a staggered Wilson kernel show clearly superior chiral properties in the Schwinger
4
Staggered domain wall fermions Christian Hoelbling
0
0.5
1
1.5
2
2.5
3
Re(λ)
-1.5
-1
-0.5
0
0.5
1
1.5
Im(λ)
8.33×10
-2
2.73×10
-1
5.39×10
-1
1.10×10
0
2.78×10
-2
3.15×10
-2
∆
N
∆
GW
std
Bor
opt
(a) Wilson kernel
0
0.5
1
1.5
2
2.5
3
Re(λ)
-1.5
-1
-0.5
0
0.5
1
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
∆
N
∆
GW
std
Bor
opt
(b) Staggered Wilson kernel
Figure 4:
Spectrum of
ρD
eff
with a Wilson kernel and a staggered Wilson kernel at
N
s
= 8
in QCD
4
at
β = 6
.
model in the limit of large
β
, where in some cases chiral symmetry violations are reduced by more
than an order of magnitude.
Quenched quantum chromodynamics.
Going beyond our study in Ref. [
16
], we finally discuss
first results for staggered domain wall fermions in the setting of quenched quantum chromodynamics
in four dimensions. In Fig. 4, we show the eigenvalue spectrum of the standard, Boriçi’s and the
optimal construction at
N
s
= 8
on a smeared
6
4
configuration at
β = 6
. The configuration was
smeared with one APE smearing iteration using
α = 0.65
. For Wilson fermions we use a domain
wall height of
M
0
= 1.4
, while for staggered Wilson fermions
M
0
= 1
remains the canonical choice.
As we are dealing with a topologically trivial configuration, we omit m
eff
in the figure labels.
Although it is too early for a full assessment of the chiral properties of the effective operators
without obtaining more statistics, we note that on the present gauge configuration we observe a large
reduction of
∆
N
and
∆
GW
by typically more than an order of magnitude for Boriçi’s and the optimal
construction when using the staggered Wilson kernel. For the standard construction, however, the
Wilson-based construction shows better chiral properties.
4. Conclusions
The use of staggered domain wall fermions results in significantly improved chiral properties
in the Schwinger model, offering the prospect of a computationally cheaper construction. In the
setting of four-dimensional quantum chromodynamics, our first results are also very encouraging
and warrant further investigations.
5
Staggered domain wall fermions Christian Hoelbling
Acknowledgments.
C. H. is supported by DFG grant SFB/TRR-55. C. Z. is supported by the
Singapore International Graduate Award (SINGA) and Nanyang Technological University.
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