Spectral properties and chiral symmetry violations
of (staggered) domain wall fermions in the Schwinger model
Christian Hoelbling
Department of Physics, University of Wuppertal, D-42119 Wuppertal, Germany
Christian Zielinski
Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371 and
Department of Physics, University of Wuppertal, D-42119 Wuppertal, Germany
We follow up on a suggestion by Adams and construct explicit domain wall fermion operators
with staggered kernels. We compare different domain wall formulations, namely the standard con-
struction as well as Boriçi’s modified and Chiu’s optimal construction, utilizing both Wilson and
staggered kernels. In the process, we generalize the staggered kernels to arbitrary even dimensions
and introduce both truncated and optimal staggered domain wall fermions. Some numerical in-
vestigations are carried out in the (1 + 1)-dimensional setting of the Schwinger model, where we
explore spectral properties of the bulk, effective and overlap Dirac operators in the free-field case,
on quenched thermalized gauge configurations and on smooth topological configurations. We com-
pare different formulations using the effective mass, deviations from normality and violations of the
Ginsparg-Wilson relation as measures of chirality.
PACS numbers: 11.15.Ha, 71.10.Fd
Keywords: Domain wall fermions, staggered Wilson fermions, Schwinger model, chiral symmetry
CONTENTS
I. Introduction 1
II. Kernel operators 2
A. Wilson kernel 2
B. Staggered Wilson fermions 2
III. Domain wall fermions 3
A. Standard construction 3
B. Boriçi’s construction 4
C. Optimal construction 4
D. Staggered formulations 5
IV. Effective Dirac operator 5
A. Derivation 6
B. The N
s
→ ∞ limit 6
C. Approximate sign functions 7
V. Setting 7
A. Choice of M
0
7
B. Effective mass 7
C. Normality and Ginsparg-Wilson relation 8
D. Topological charge 8
VI. Numerical results 8
A. Free-field case 8
B. U(1) gauge field case 10
C. Smearing 13
D. Topology 13
E. Spectral flow 16
F. Approaching the continuum 17
VII. Conclusions 20
Acknowledgments 20
A. Optimal weights 20
References 20
I. INTRODUCTION
Chiral symmetry plays a crucial role in the understand-
ing of hadron phenomenology and the low-energy dynam-
ics of quantum chromodynamics (QCD). On the lattice,
the overlap construction [1–7] allows one to implement a
fermion operator with exact chiral symmetry [8–10], thus
evading the Nielsen-Ninomiya theorem [11–14]. In prac-
tice, the use of overlap fermions is limited by the fact
that they generically require a factor of O(10–100) more
computational resources than Wilson fermions and tun-
neling between topological sectors is severely suppressed
even at moderate lattice spacings [15–18].
Domain wall fermions [19–21] offer an alternative by
formulating fermions with approximate chiral symmetry
in d dimensions by means of massive interacting fermions
in d + 1 dimensions (d = 2, 4). The limit of an infinite
extension of the extra dimension can again be expressed
as an overlap operator with exact chiral symmetry. For
a finite extent, domain wall fermions can then be seen as
a truncation of overlap fermions. They offer the possi-
bility of reducing computational cost and are well suited
for parallel implementations. This comes at the price of
replacing the exact chiral symmetry by an approximate
one. It is expected that chiral symmetry violations are
exponentially suppressed [20, 22–24], although in prac-
tice this suppression can still require large extents of the
extra dimension [25–29]. However, these violations also
facilitate the tunneling between topological sectors.
Domain wall fermions are typically formulated with
a Wilson kernel [30]. Only recently has it been clari-
fied by Adams [31, 32] how to utilize the computation-
ally more efficient staggered fermions [33–36] in its place
by giving staggered fermions a flavor-dependent mass;
see also Refs. [37–39]. Subsequent numerical work [40–
arXiv:1602.08432v2 [hep-lat] 5 Jul 2016
2
43] focused on the properties of these staggered Wilson
fermions and their use as a kernel for an overlap con-
struction [32]. The possibility of staggered domain wall
fermions, which was also suggested in Ref. [32], has how-
ever not been investigated any further. The present work
is meant as a first step in closing this gap. We give ex-
plicit constructions of staggered domain wall fermions
and compare their spectral and chiral symmetry breaking
properties to those of traditional domain wall fermions
with Wilson fermions in the context of the Schwinger
model [44].
While we are eventually interested in QCD, the
Schwinger model, i.e. (1 + 1)-dimensional quantum elec-
trodynamics (QED), retains enough properties of QCD.
In particular, we find confinement and topological struc-
ture, making it useful for conceptual investigations. On
the other hand, it is numerically simple enough to al-
low the computation of the complete eigenvalue spectrum
of fermion operators on nontrivial background configura-
tions. Moreover, the study of fermions in 1+1 dimensions
naturally arises e.g. in the low-energy description of con-
ducting electrons in metals, see Ref. [45].
This paper is organized as follows. In Sec. II, we dis-
cuss the kernel operators, among them generalizations of
staggered Wilson fermions in an arbitrary even number
of dimensions. In Sec. III, the construction of (staggered)
domain wall fermions and their variations are given, in
Sec. IV we introduce the effective Dirac operators and
discuss the limiting overlap operators, in Sec. V we ex-
plain our approach of carrying out the numerical calcu-
lations and in Sec. VI we discuss the numerical results.
In Sec. VII, we conclude our work and give an outlook.
II. KERNEL OPERATORS
We begin by giving a quick review of the kernel op-
erators we are considering, namely Wilson and stag-
gered Wilson fermions. Here and in the following, we
are mostly interested in the (1 + 1)-dimensional case
(d = 2), but where convenient we write down the general
d-dimensional expressions.
A. Wilson kernel
For Wilson fermions [46], the Dirac operator reads
D
w
(m
f
) = γ
µ
∇
µ
+ m
f
+ W
w
. (1)
Here the γ
µ
matrices refer to a representation of the Dirac
algebra {γ
µ
, γ
ν
} = 2δ
µ,ν
1
with µ ∈ {1, . . . , d}, δ
µ,ν
to
the Kronecker delta, ∇
µ
to the covariant central finite
difference operator and m
f
to the bare fermion mass. The
Wilson term reads
W
w
= −
ar
2
∆ (2)
with lattice spacing a, Wilson parameter r ∈ (0, 1] and
the covariant lattice Laplacian ∆. We note that D
†
w
=
γ
d+1
D
w
γ
d+1
, where γ
d+1
is the chirality matrix, and that
the W
w
term breaks chiral symmetry explicitly. In terms
of the parallel transport
T
µ
Ψ (x) = U
µ
(x) Ψ (x + aˆµ) (3)
we have the following definitions:
∇
µ
=
1
2a
T
µ
− T
†
µ
, (4)
C
µ
=
1
2
T
µ
+ T
†
µ
, (5)
∆ =
2
a
2
X
µ
(C
µ
−
1
) . (6)
Through the Wilson term W
w
, the doublers acquire a
mass O
a
−1
. In the continuum limit, the number of
flavors is then reduced from 2
d
to one physical flavor.
B. Staggered Wilson fermions
Following Refs. [31, 32, 37, 38, 41, 42], in d = 4 dimen-
sions a staggered Wilson operator can be written as
D
sw
(m
f
) = D
st
+ m
f
+ W
st
(7)
with staggered Dirac operator
D
st
= η
µ
∇
µ
, (8)
η
µ
χ (x) = (−1)
P
ν<µ
x
ν
/a
χ (x) (9)
and bare fermion mass m
f
. The staggered Wilson term is
an operator that, up to discretization terms, is trivial in
spin but splits the different flavors. We also require the
Dirac operator to have a real determinant. The original
suggestion by Adams [31, 32] reads
W
st
=
r
a
(
1
+ Γ
1234
C
1234
) (10)
with a Wilson-like parameter r > 0 and operators
Γ
1234
χ (x) = (−1)
P
µ
x
µ
/a
χ (x) , (11)
C
1234
= η
1
η
2
η
3
η
4
(C
1
C
2
C
3
C
4
)
sym
. (12)
In the spin-flavor basis, this term has the form
W
st
∼
r
a
1
⊗ (
1
− ξ
5
) + O(a) , (13)
which splits the four flavors into pairs of two according to
their “flavor chirality”, i.e. the eigenbasis of ξ
5
. The nota-
tion A ∼ B means that B corresponds to the respective
spin ⊗flavor interpretation [37] of A up to proportional-
ity, while the ξ
µ
are a representation of the Dirac algebra
in flavor space. The determinant of D
sw
is real due to
the -Hermiticity
D
†
sw
= D
sw
(14)
3
with
(x) = (−1)
P
µ
x
µ
/a
. (15)
In four dimensions, one may also split the flavors with
respect to the eigenbasis of different elements of the flavor
Dirac algebra [37, 38]. To retain Hermiticity and, thus,
a real determinant, the flavor structure of the mass term
needs to be restricted to a sum of products of an even
number of ξ
µ
. A single flavor staggered fermion in four
dimensions can, thus, be obtained by e.g.
W
st
=
r
a
2 ·
1
+ W
12
st
+ W
34
st
, (16)
W
µν
st
= i Γ
µν
C
µν
, (17)
where the operators Γ
µν
and C
µν
are given by
Γ
µν
χ (x) = ε
µν
(−1)
(x
µ
+x
ν
)/a
χ (x) , (18)
C
µν
= η
µ
η
ν
·
1
2
(C
µ
C
ν
+ C
ν
C
µ
) (no sum) . (19)
Here and in the following, ε
µ
1
···µ
N
refers to the Levi-
Civita symbol. To interpret the mass term in Eq. (17),
note that
Γ
µν
∼ γ
µ
γ
ν
⊗ ξ
µ
ξ
ν
, (20)
C
µν
∼ γ
µ
γ
ν
⊗
1
, (21)
∼ γ
5
⊗ ξ
5
, (22)
up to discretization terms. As a result, we find for the
staggered Wilson term
W
µν
st
∼
1
⊗ σ
µν
+ O(a) (23)
with σ
µν
= i ξ
µ
ξ
ν
. This implies that the number of phys-
ical fermion species of W
st
is reduced to one by giving all
but a single flavor a mass O
a
−1
.
These results can be formulated in an arbitrary even
number of dimensions d, where we can write a single
flavor mass term as
W
st
=
r
a
d/2
X
k=1
1
+ W
(2k−1)(2k)
st
(24)
and Eq. (16) now follows as the special case d = 4. To
construct a more general mass term, we define
W
µ
1
···µ
2n
st
= i
n
Γ
µ
1
···µ
2n
C
µ
1
···µ
2n
, (25)
for an arbitrary n ≤ d/2, where
Γ
µ
1
···µ
2n
χ (x) = ε
µ
1
···µ
2n
(−1)
P
2n
i=1
x
µ
i
/a
χ (x) , (26)
C
µ
1
···µ
2n
= η
µ
1
···η
µ
2n
(C
µ
1
···C
µ
2n
)
sym
. (27)
In the spin ⊗flavor interpretation, we find
Γ
µ
1
···µ
2n
∼ (γ
µ
1
. . . γ
µ
2n
) ⊗ (ξ
µ
1
. . . ξ
µ
2n
) , (28)
C
µ
1
···µ
2n
∼ (γ
µ
1
. . . γ
µ
2n
) ⊗
1
, (29)
W
µ
1
···µ
2n
st
∼
1
⊗ (i
n
ξ
µ
1
. . . ξ
µ
2n
) , (30)
up to discretization terms. In addition, the new mass
terms fulfill the -Hermiticity relation
W
†
= W , W ≡ W
µ
1
···µ
2n
st
. (31)
We can, thus, replace Eq. (24) by a generic
W
st
=
d/2
X
n=1
X
µ
n
r
µ
n
a
1
+ W
µ
n
st
, (32)
where r
µ
n
≥ 0 and the sum is over all multi-indices µ
n
=
(µ
1
, . . . , µ
2n
) with 1 ≤ µ
i
≤ d for all i with 1 ≤ i ≤ 2n
[47]. Adams’ original mass term in d = 4 dimensions in
Eq. (10) then follows from setting r
1234
= r > 0 and
r
µ
n
= 0 otherwise.
For the d = 2 case that we will consider in the numeri-
cal part of this paper, the definition is essentially unique
and reads
W
st
=
r
a
1
+ W
12
st
. (33)
In this case the reduction is from two staggered flavors
to a single physical one.
Like in the Wilson case, all possible W
st
terms break
chiral symmetry explicitly. Furthermore, there may be
additional counterterms if too many of the staggered
symmetries are broken [39].
III. DOMAIN WALL FERMIONS
After having introduced the kernel operators, we now
move on to the domain wall fermion Dirac operators.
Originally proposed by Kaplan [19], then refined by
Shamir and Furman [20, 21], the domain wall construc-
tion implements approximately massless fermions in d
dimensions by means of a (d + 1)-dimensional theory.
Equivalently, domain wall fermions can be understood as
a tower of N
s
fermions in d dimensions with a particular
flavor structure.
We now give a quick summary of the well-known
(d + 1)-dimensional formulations. For the remainder of
the paper we fix the d-dimensional lattice spacing to
a = 1 and the (staggered) Wilson parameter to r = 1.
A. Standard construction
We begin with the standard construction. First, let us
define
D
±
w
= a
d+1
D
w
(−M
0
) ±
1
, (34)
where we explicitly write out the lattice spacing a
d+1
in
the extra dimension. The parameter M
0
is the so-called
domain wall height and must be suitably chosen for the
4
description of a single flavor. In the free-field case, we
have M
0
∈ (0, 2r).
The Dirac operator reads
ΨD
dw
Ψ =
N
s
X
s=1
Ψ
s
D
+
w
Ψ
s
− P
−
Ψ
s+1
− P
+
Ψ
s−1
(35)
with (d + 1)-dimensional fermion fields Ψ, Ψ and chiral
projectors P
±
=
1
2
(
1
± γ
d+1
). Here and in the following,
the index s refers to the additional spatial (or equiva-
lently flavor) coordinate. The gauge links are taken to
be the identity matrix along the additional coordinate.
Furthermore, we impose the following boundary condi-
tions,
P
+
(Ψ
0
+ mΨ
N
s
) = 0, (36)
P
−
(Ψ
N
s
+1
+ mΨ
1
) = 0, (37)
where m is related to the bare fermion mass, see Eq. (76).
We note that in the special case of m = 0 we find Dirich-
let boundary conditions in the extra dimension, while
for m = ±1 one recovers (anti-)periodic boundary condi-
tions. If we write down the Dirac operator in the extra
dimension explicitly, we find
D
dw
=
D
+
w
−P
−
mP
+
−P
+
D
+
w
−P
−
.
.
.
.
.
.
.
.
.
−P
+
D
+
w
−P
−
mP
−
−P
+
D
+
w
. (38)
One possibility of constructing the d-dimensional fermion
fields from the boundary is via
q = P
+
Ψ
N
s
+ P
−
Ψ
1
, q = Ψ
1
P
+
+ Ψ
N
s
P
−
. (39)
Let us also define the reflection operator along the ex-
tra dimension
R =
1
.
.
.
1
. (40)
We find that D
dw
is Rγ
d+1
-Hermitian
D
†
dw
= Rγ
d+1
· D
dw
· Rγ
d+1
, (41)
which ensures that det D
dw
∈ R and the applicability of
importance sampling techniques.
Besides this canonical formulation, several variations
of domain wall fermions have been proposed.
B. Boriçi’s construction
One of them is the construction by Boriçi [48], which
follows from the original proposal by the replacements
P
+
Ψ
s−1
→ −D
−
w
P
+
Ψ
s−1
, (42)
P
−
Ψ
s+1
→ −D
−
w
P
−
Ψ
s+1
. (43)
Note that this is an O(a
d+1
) modification. The Dirac
operator in its full form reads
ΨD
dw
Ψ =
N
s
X
s=1
Ψ
s
D
+
w
Ψ
s
+ D
−
w
P
−
Ψ
s+1
+ D
−
w
P
+
Ψ
s−1
(44)
or explicitly
D
dw
=
D
+
w
D
−
w
P
−
−mD
−
w
P
+
D
−
w
P
+
D
+
w
D
−
w
P
−
.
.
.
.
.
.
.
.
.
D
−
w
P
+
D
+
w
D
−
w
P
−
−mD
−
w
P
−
D
−
w
P
+
D
+
w
.
(45)
Furthermore, Eq. (39) generalizes to
q = P
+
Ψ
N
s
+ P
−
Ψ
1
, (46)
q = −Ψ
1
D
−
w
P
+
− Ψ
N
s
D
−
w
P
−
(47)
and Eq. (41) to
D
−1
D
dw
†
= Rγ
d+1
·
D
−1
D
dw
· Rγ
d+1
, (48)
D =
1
N
s
⊗ D
−
w
(49)
(see Ref. [49]). This formulation is also known as
“truncated overlap fermions” as the corresponding d-
dimensional effective operator equals the polar decompo-
sition approximation [6, 50] of order N
s
/2 of Neuberger’s
overlap operator (for even N
s
).
C. Optimal construction
The last modification we consider are the optimal do-
main wall fermions proposed by Chiu [51, 52]. The idea
is to modify D
dw
in such a way, that the effective Dirac
operator is expressed through Zolotarev’s optimal ratio-
nal approximation of the sign function [53–55] (see also
Refs. [56, 57]). In the following, we quote the central
formulas of the construction given in Ref. [51].
Starting from Boriçi’s construction, the Dirac operator
is modified by introducing weight factors
ΨD
dw
Ψ =
N
s
X
s=1
Ψ
s
D
+
w
(s) Ψ
s
+
N
s
X
s=1
Ψ
s
D
−
w
(s) P
−
Ψ
s+1
+ D
−
w
(s) P
+
Ψ
s−1
, (50)
where
D
±
w
(s) = a
d+1
ω
s
D
w
(−M
0
) ±
1
. (51)
5
The weight factors ω
s
are given by
ω
s
=
1
λ
min
p
1 − κ
02
sn (v
s
, κ
0
) (52)
with sn (v
s
, κ
0
) being the corresponding Jacobi elliptic
function with argument v
s
and modulus κ
0
. The modulus
is defined by
κ
0
=
q
1 − λ
2
min
/λ
2
max
(53)
and λ
2
min
(λ
2
max
) is the respective smallest (largest) eigen-
value of H
2
w
with
H
w
= γ
d+1
D
w
(−M
0
) . (54)
The argument v
s
reads
v
s
= (−1)
s−1
M sn
−1
s
1 + 3λ
(1 + λ)
3
,
p
1 − λ
2
!
+
j
s
2
k
2K
0
N
s
,
(55)
where
λ =
N
s
Y
`=1
Θ
2
(2`K
0
/N
s
, κ
0
)
Θ
2
((2` − 1) K
0
/N
s
, κ
0
)
, (56)
M =
bN
s
/2c
Y
`=1
sn
2
((2` − 1) K
0
/N
s
, κ
0
)
sn
2
(2`K
0
/N
s
, κ
0
)
. (57)
Here b·c refers to the floor function, K
0
= K (κ
0
) to the
complete elliptic integral of the first kind with
K (k) =
π/2
ˆ
0
dθ
p
1 − k
2
sin
2
θ
. (58)
Furthermore we introduced the elliptic Theta function
via
Θ (w, k) = ϑ
4
πw
2K
, k
(59)
with K = K (k) and elliptic theta functions ϑ
i
[58]. Some
reference values for the weight factors {ω
s
} can be found
in App. A.
Similarly to Boriçi’s construction, Eq. (39) now gener-
alizes to
q = P
+
Ψ
N
s
+ P
−
Ψ
1
, (60)
q = −Ψ
1
D
−
w
(1) P
+
− Ψ
N
s
D
−
w
(N
s
) P
−
(61)
and Eq. (41) again to Eq. (48), but now with
D = diag
D
−
w
(1) , . . . , D
−
w
(N
s
)
, (62)
as pointed out in Ref. [49]. Optimal domain wall fermions
have been and are still extensively used. Some of the
results obtained can be found in Refs. [59–64].
We also note that there is a modified construction of
optimal domain wall fermions [65], which is reflection-
symmetric along the fifth dimension. For completeness
we point out that all the preceding domain wall fermion
formulations can be seen as special cases of Möbius do-
main wall fermions [49, 66, 67].
D. Staggered formulations
As proposed in Ref. [32], we can use the staggered Wil-
son kernel to formulate a staggered version of domain wall
fermions. We can write the Dirac operator in a general
d-dimensional form as
ΥD
sdw
Υ =
N
s
X
s=1
Υ
s
D
+
sw
Υ
s
− P
−
Υ
s+1
− P
+
Υ
s−1
, (63)
where Υ refers to the staggered fermion field. Like in the
Wilson case we define
D
±
sw
= a
d+1
D
sw
(−M
0
) ±
1
. (64)
The chiral projectors are given by P
±
=
1
2
(
1
± ), where
2
=
1
. Here we have ∼ γ
d+1
⊗ ξ
d+1
, which reduces to
∼ γ
d+1
⊗
1
on the physical species. One can easily verify
the R-Hermiticity of D
sdw
. Note that we follow a sign
convention in where our D
sdw
is in full analogy to D
dw
,
while in Ref. [32] a slightly different convention is used.
The staggered domain wall Dirac operator D
sdw
can be
constructed from D
dw
by the replacement rule given in
Ref. [32], which we write down in a general d-dimensional
form as
γ
d+1
→ , D
w
→ D
sw
. (65)
Using the replacement rule in Eq. (65), we can also
generalize Boriçi’s and the optimal construction to the
case of a staggered Wilson kernel. This gives rise to pre-
viously not considered truncated staggered domain wall
fermions with the Dirac operator
ΥD
sdw
Υ =
N
s
X
s=1
Υ
s
D
+
sw
Υ
s
+
N
s
X
s=1
Υ
s
D
−
sw
P
−
Υ
s+1
+ D
−
sw
P
+
Υ
s−1
(66)
as well as optimal staggered domain wall fermions
ΥD
sdw
Υ =
N
s
X
s=1
Υ
s
D
+
sw
(s) Υ
s
+
N
s
X
s=1
Υ
s
D
−
sw
(s) P
−
Υ
s+1
+ D
−
sw
(s) P
+
Υ
s−1
, (67)
where D
±
sw
(s) = a
d+1
ω
s
D
sw
(−M
0
) ±
1
and the weight
factors ω
s
are given by Eq. (52) for the kernel H
sw
=
D
sw
(−M
0
).
IV. EFFECTIVE DIRAC OPERATOR
To understand the relation between the (d + 1)-
dimensional fermions and the light d-dimensional fields
q, q at the boundary, we introduce the effective d-
dimensional Dirac operator as derived in Refs. [48, 68–70]
(see also Refs. [71, 72]). In the following, we give a short
summary, following Refs. [48, 69, 72].
6
A. Derivation
The low energy effective d-dimensional action
S
eff
=
X
x
q (x) D
eff
q (x) (68)
follows after integrating out the N
s
−1 heavy modes. The
effective Dirac operator is defined via the propagator
D
−1
eff
(x, y) = hq (x) q (y)i. (69)
For a suitable choice of M
0
, there is exactly one light and
N
s
− 1 heavy Dirac fermions.
In the chiral limit N
s
→ ∞ (at fixed bare coupling β),
the contribution from the heavy fermions diverges. This
bulk contribution from the (d + 1)-dimensional fermions
can be canceled by the introduction of suitable pseudo-
fermion fields. One typically chooses the fermion action
with the replacement m → 1 as the action for the pseudo-
fermions.
Let us begin by defining the Hermitian operators
H
w
= γ
d+1
D
w
(−M
0
) , (70)
H
m
= γ
d+1
D
m
(−M
0
) , (71)
where the kernel operator of standard domain wall
fermions is given by
D
m
(−M
0
) =
D
w
(−M
0
)
2 ·
1
+ a
d+1
D
w
(−M
0
)
. (72)
The transfer matrix along the extra dimension is given
by
T =
T
−
T
+
, T
±
=
1
± a
d+1
H, (73)
where we use the notation
H =
(
H
m
for standard constr.,
H
w
for Boriçi’s constr.
(74)
Then the effective operator can be written 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
−
. (75)
Note that we can rewrite Eq. (75) as
D
eff
= (1 − m)
D
eff
(0) +
m
1 − m
, (76)
where D
eff
(0) denotes the effective operator D
eff
at m =
0. We can see that the parameter m induces a bare
fermion mass of m/ (1 − m), see Ref. [73].
Alternatively, one can also show [48, 72] the relation
D
eff
=
P
|
D
−1
1
D
m
P
1,1
(77)
with the matrix P defined as
P =
P
−
P
+
P
−
P
+
.
.
.
.
.
.
P
−
P
+
P
+
P
−
(78)
and P
−1
= P
|
. Here we used the shorthand notation
D
m
= D
dw
(m), while the index stands for the (1, 1) s-
block of the matrix.
The derivation of the effective operator for optimal do-
main wall fermions follows Boriçi’s construction after in-
cluding the weight factors {ω
s
} appropriately [51]. By
construction the sign-function approximation equals the
optimal rational approximation. It can be either evalu-
ated directly or via the projection method of Eq. (77).
B. The N
s
→ ∞ limit
In the following, let us specialize to m = 0. Note that
we can rewrite Eq. (75) using
T
N
s
+
− T
N
s
−
T
N
s
+
+ T
N
s
−
=
N
s
/2
(a
d+1
H) , (79)
where
N
s
/2
is Neuberger’s polar decomposition approxi-
mation [6, 50] of the sign-function. Therefore, we obtain
an overlap operator in the N
s
→ ∞ limit as follows,
D
ov
= lim
N
s
→∞
D
eff
=
1
2
1
+
1
2
γ
d+1
sign H
=
1
2
1
+ D
−M
0
D
†
−M
0
D
−M
0
−
1
2
, (80)
with H given in Eq. (74), D
−M
0
= D (−M
0
) and
D =
(
D
m
for standard constr.,
D
w
for Boriçi’s/Chiu’s constr.
(81)
The overlap operator satisfies the Ginsparg-Wilson equa-
tion
{γ
d+1
, D
ov
} = 2D
ov
γ
d+1
D
ov
(82)
and allows for an exact chiral symmetry. Eq. (82) also
implies the normality of the overlap operator as can be
easily verified.
Comparing Eq. (80) to the standard definition of the
overlap operator
D
ov
= ρ
1
+ D
−ρ
D
†
−ρ
D
−ρ
−
1
2
(83)
and using the relation for the effective negative mass pa-
rameter
ρ =
(
M
0
−
a
d+1
2
M
2
0
for standard constr.,
M
0
for Boriçi’s/Chiu’s constr.,
(84)
7
we would obtain a restriction on the domain wall height
M
0
from ρ = 1/2. This can be avoided by simply rescal-
ing D
eff
by a factor % = 2ρ, so that—up to discretization
effects—the low-lying eigenvalues of the kernel remain
invariant under the effective operator projection in the
free-field case. This is also illustrated in Fig. 8, which we
elaborate on in Sec. VI A. Consequently, we will employ
this rescaling in all our numerical investigations.
C. Approximate sign functions
The effective Dirac operators in the various formula-
tions are given in terms of different sign function approxi-
mations. Explicitly, these approximations of sign (z) read
r (z) =
Π
+
(z) − Π
−
(z)
Π
+
(z) + Π
−
(z)
(85)
with
Π
±
(z) =
(
(1 ± z)
N
s
for standard/Boriçi’s constr.,
Q
s
(1 ± ω
s
z) for optimal constr.,
(86)
so that r (z) → sign (z) for N
s
→ ∞. We illustrate these
approximations in Fig. 1, comparing the polar decom-
position approximation in Boriçi’s construction with the
optimal rational function approximation in Chiu’s con-
struction. The coefficients {ω
s
} are directly linked to
Zolotarev’s coefficients, cf. Refs. [53, 56, 57]. Note that
the sign function approximation for the standard con-
struction agrees with the one in Boriçi’s construction,
but is applied to H
m
rather than H
w
.
−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
s
= 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
s
= 6
Figure 1: Approximation of sign (z) by r (z). The opti-
mal construction was done for λ
min
= 1, λ
max
= 3.
Staggered formulations. As previously suggested in
Ref. [32], all central equations in this section generalize to
the case of staggered Wilson fermions after the replace-
ments given in Eq. (65). In particular, staggered domain
wall fermions can be seen as a truncation of Neuberger’s
overlap construction with staggered Wilson kernel [32].
V. SETTING
In the following, we elaborate on the setting of our
numerical simulations. In particular, we discuss our ap-
proach of comparing the chiral properties of the different
formulations.
A. Choice of M
0
In the free-field case, suitable choices for the domain
wall height M
0
are in the range 0 < M
0
< 2 for a one
flavor theory. In a gauge field background, in general this
interval contracts. Intuitively, the parameter M
0
has to
be chosen in such a way, that the origin is shifted suffi-
ciently close to the center of the leftmost “belly” of the
eigenvalue spectrum. While there is no unique optimal
choice even in the free-field case [74], one can use the
canonical choice M
0
= 1, which shifts the origin exactly
to the center. To be more precise, the mobility edge [75–
78] is the decisive quantity determining valid choices of
M
0
.
In the Schwinger model, the interval of valid choices
for M
0
remains close to the free-field case for reasonable
values of the inverse coupling β. In particular, M
0
= 1
remains a sensible and simple choice, which we are con-
sequently using for all numerical work presented. This
is in contrast to QCD in 3 + 1 dimensions, where one
commonly sets M
0
= 1.8 (see e.g. Ref. [79]).
B. Effective mass
Two common approaches found in the literature to
quantify the induced effects of chiral symmetry break-
ing in domain wall fermions are the determination of the
residual mass m
res
[21, 27, 79, 80] and the effective mass
m
eff
[81–83]. The former employs the explicit fermion
mass dependence in the chiral Ward-Takahashi identi-
ties, while the latter is given by the lowest eigenvalue of
the Hermitian operator in a topologically nontrivial back-
ground field. Although the definitions are not equivalent,
their numerical values usually agree within a factor of
O(1) and hence both are suitable to quantify the degree
of chiral symmetry breaking.
In this work, we use the effective mass m
eff
due to its
conceptual simplicity. We hence define the effective mass
for a given Dirac operator D with periodic boundary con-
ditions in a topologically nontrivial background field as
the lowest eigenvalue of the corresponding Hermitian ver-
sion H of the Dirac operator. Noting that H
2
= D
†
D,
we define
m
eff
= min
λ∈spec H
|λ| = min
Λ∈spec D
†
D
√
Λ. (87)
8
If D is a normal operator, then m
eff
= min
λ∈spec D
|λ|. In
the general case, however, there is no direct link between
the eigenvalues of H and D.
On a topological nontrivial background configuration
with topological charge Q 6= 0, the Atiyah-Singer index
theorem [84–87] ensures the existence of zero modes of H
in the continuum. The corresponding lattice version of
this theorem [8–10] ensures that the overlap operator in
Eq. (80) has exact zero modes as well. For domain wall
fermions and their respective effective operators these
zero modes are recovered in the N
s
→ ∞ limit. For finite
N
s
, however, these zero modes become approximate and
their deviation from zero can serve as a measure for the
degree of chiral symmetry breaking.
If n
∓
refers to the number of left-handed and right-
handed zero modes, then the Atiyah-Singer index theo-
rem states that
n
−
− n
+
= (−1)
d/2
Q, (88)
see Ref. [31]. A precise definition of Q will be given in
Eq. (92). We note here that in 1 + 1 dimensions the
Vanishing Theorem holds [88–90]. That is, if Q 6= 0,
then either n
−
or n
+
vanishes.
C. Normality and Ginsparg-Wilson relation
In the continuum, the Dirac operator is normal. The
same holds for the naïve and staggered discretizations as
well as for the overlap operator. The (staggered) Wilson
kernel and the (staggered) domain wall fermion opera-
tors, on the other hand, are not normal.
As it has been shown that normality is necessary for
chiral properties [91] (see also Refs. [92, 93]), the degree
of violation of normality is an interesting quantity in the
context of chiral symmetry.
Let us recall that a normal operator D satisfies
D, D
†
= 0 by definition. We then consider the quantity
∆
N
=
D, D
†
∞
, (89)
where k·k
∞
is the by the L
∞
-norm induced matrix norm.
We know that ∆
N
has to vanish for the effective operators
introduced in Sec. IV in the limit N
s
→ ∞.
Similarly, we consider violations of the Ginsparg-
Wilson relation given in Eq. (82). The quantity
∆
GW
=
{γ
3
, D} − ρ
−1
Dγ
3
D
∞
, (90)
has to vanish in the limit N
s
→ ∞ as well. As before,
we replace γ
3
by in the case of a staggered Wilson ker-
nel. As previously already considered in Refs. [74, 94],
∆
N
and ∆
GW
will give us a measure for the degree of
chiral symmetry violation of the Dirac operators under
consideration.
D. Topological charge
We determine the topological charge of the gauge con-
figurations via both the standard overlap definition,
Q =
1
2
Tr
H
w
/
p
H
2
w
, (91)
and its staggered counterpart,
Q =
1
2
Tr
H
sw
/
p
H
2
sw
, (92)
with H
sw
= D
sw
(−M
0
) as derived in Ref. [31]. On the
small sample of gauge configurations considered in this
paper, they were found to be in exact agreement. Al-
though a more careful investigation of the continuum
limit would be needed, this observation is consistent
with analytical results [95] and other numerical studies
[41, 96].
VI. NUMERICAL RESULTS
We are now moving to the numerical part of this work.
We calculate the complete eigenvalue spectra of all Dirac
operators introduced in the previous sections, both with
a Wilson and staggered Wilson kernel, for the (1 + 1)-
dimensional Schwinger model. We consider the free-field
case, thermalized gauge configurations and the smooth
topological configurations constructed in Ref. [97].
In the following, we set the lattice spacing to a =
a
d+1
= 1 and Wilson parameter to r = 1. The ex-
tent in the extra dimension will be varied in the range
2 ≤ N
s
≤ 8. We use periodic boundary conditions in
both space and time direction, so that the determination
of the effective mass m
eff
as defined in Sec. V B applies.
Extremal eigenvalues are determined with arpack
[98], while complete spectra are computed with lapack
[99]. Calculations are carried out in double precision. In
all figures, the abbreviation “std” refers to the standard
construction, “Bor” to Boriçi’s construction and “opt” to
Chiu’s optimal construction. With respect to the over-
lap constructions, “DW” refers to the overlap operator
with kernel H
m
, “Neub” to Neuberger’s overlap with ker-
nel H
w
and “Adams” to Adams’ staggered overlap with
kernel H
sw
.
A. Free-field case
We begin with the free-field case. Here we can employ
a momentum space representation of the kernel. In par-
ticular, the Wilson kernel can be represented as a 2 × 2
linear map
D
w
= (m
f
+ 2r)
1
+ i
X
µ
γ
µ
sin p
µ
− r
X
ν
cos p
ν
1
, (93)
9
0
0.5
1
1.5
2
2.5
3
3.5
4
Re(λ)
-1.5
-1
-0.5
0
0.5
1
1.5
Im(λ)
Wilson
0
0.5
1
1.5
2
2.5
3
3.5
4
Re(λ)
-1.5
-1
-0.5
0
0.5
1
1.5
Im(λ)
staggered Wilson
Figure 2: Free-field spectrum of kernel operators.
where p
µ
= 2πn
µ
/N
µ
with n
µ
= 0, 1, . . . , N
µ
− 1 and
N
µ
is the number of slices in µ-direction. The staggered
Wilson kernel takes the form of the 4 × 4 linear map
D
sw
= m
f
(
1
⊗
1
) + i
X
µ
sin p
µ
(γ
µ
⊗
1
)
+ r
1
⊗
1
+ ξ
3
Y
ν
cos p
ν
!
(94)
with n
µ
= 0, 1, . . . , N
µ
/2 − 1 (see also Refs. [41, 95]).
In the three-dimensional operators, we keep the extra
dimension in the position space formulation, as in general
it is lacking periodicity. Besides reducing the dimension-
ality of the eigenvalue problem by choosing a momentum
space representation for the kernel, this also avoids some
numerical instabilities of the free-field case encountered
in a purely position space based formulation.
In the following, all numerical results are for the case
of a N
s
× N
t
= 20 × 20 lattice.
a. Kernel operators. In Fig. 2, we can find the well
known free spectra of the kernel operators. In 1 + 1 di-
mensions the Wilson Dirac operator has only two doubler
branches in the eigenvalue spectrum due to the reduced
number of fermion species in this low-dimensional set-
ting. Like in 3 + 1 dimensions, the staggered Wilson
Dirac operator has a single doubler branch due to the
splitting of positive and negative flavor-chirality species.
One can see that the free-field spectrum of the stag-
gered Wilson kernel is closer to the Ginsparg-Wilson cir-
cle compared to the Wilson kernel. One can then hope
for a better performance of chiral formulations with this
kernel, at least on sufficiently smooth configurations.
The bulk and effective operators, which we are going
to discuss in the following, use either a Wilson or a stag-
gered Wilson kernel. Comparing both cases, we note that
the spectra of these operators differ mostly due to the dif-
ferent ultraviolet parts of the respective kernel spectra.
Although the low-lying parts of the kernel spectrum in
the physical branch are alike, the ultraviolet modes will
alter the resulting spectrum of the bulk and effective op-
erators differently and have an impact on the efficiency
and chiral properties.
b. Bulk operators. In Figs. 3 and 4, we show the
spectrum of the (2 + 1)-dimensional bulk operator in
the standard, Boriçi’s and the optimal construction. In
Fig. 5, we show periodic (m = −1) and antiperiodic
(m = 1) boundary conditions in the extra dimension to
compare with the Dirichlet (m = 0) case.
We can observe that the bulk spectra for Boriçi’s and
the optimal construction have lost their resemblance to a
Wilson operator in three dimensions. Specifically, while
for the standard construction we find (two) three doubler
branches in the spectrum with the (staggered) Wilson
kernel, in Boriçi’s construction one less branch is visible.
In the standard and Boriçi’s construction, we find 2N
s
exact zero modes in the Dirichlet case [20, 81], which dis-
appear for m 6= 0. In the optimal construction, we notice
how the corresponding eigenvalues get spread out along
the real axis and we are left with only two approximate
zero modes.
c. Effective operators. We now move on to the ef-
fective operators %D
eff
as defined in Eq. (75). In Figs. 6
and 7, we show the respective eigenvalue spectra with a
Wilson and staggered Wilson kernel.
As we can see, the spectra approach the Ginsparg-
Wilson circle rapidly for increasing values of N
s
. This
fast convergence is of course expected on smooth con-
figurations like the free field. Already for N
s
= 8 the
spectrum is close to the spectrum of the corresponding
overlap operator in the N
s
→ ∞ limit. In particular, we
note the rapid convergence of Boriçi’s and Chiu’s con-
struction with a staggered Wilson kernel.
The effective Dirac operator in the optimal construc-
tion shows a significantly improved convergence. Let us
recall that for a given interval I = [λ
min
, λ
max
] the opti-
mal rational function approximation r
opt
(z) of the sign
function minimizes the maximal deviation
δ
max
= max
z∈−I∪I
|sign (z) − r
opt
(z)|
= 1 ∓ r
opt
(±λ
min
) (95)
on the domain −I ∪ I. As expected, we observe that all
eigenvalues lie within a tube with diameter 2δ
max
around
the Ginsparg-Wilson circle. That is, for all eigenvalues
λ we find ||λ| − 1 | ≤ δ
max
. Noting that the sign func-
tion has a point of maximal deviation at both λ
min
, we
observe the absence of an exact zero mode in contrast to
the standard and Boriçi’s construction. However, due to
the rapid convergence of the rational function approxima-
tion, the approximate zero mode is of small magnitude
for already moderate values of N
s
.
d. Overlap operators. In the N
s
→ ∞ limit, the ef-
fective operators can be formulated as overlap operators
defined in Eq. (80) with the kernel H given in Eq. (74).
In Fig. 8, we can find the spectra of %D
ov
together with
the stereographic projection π of the eigenvalues onto the
10
-2 0 2 4
6
Re(λ)
-2
0
2
Im(λ)
(a) Standard construction
-2 0 2 4
6
Re(λ)
-2
0
2
Im(λ)
(b) Boriçi’s construction
-2 0 2 4
6
Re(λ)
-2
0
2
Im(λ)
(c) Optimal construction
Figure 3: Free-field spectrum of D
dw
with Wilson kernel for m = 0 at N
s
= 8.
-2 0 2 4
6
Re(λ)
-2
0
2
Im(λ)
(a) Standard construction
-2 0 2 4
6
Re(λ)
-2
0
2
Im(λ)
(b) Boriçi’s construction
-2 0 2 4
6
Re(λ)
-2
0
2
Im(λ)
(c) Optimal construction
Figure 4: Free-field spectrum of D
sdw
with staggered Wilson kernel for m = 0 at N
s
= 8.
-2 0 2 4
6
Re(λ)
-2
0
2
Im(λ)
(a) Periodic case (m = −1)
-2 0 2 4
6
Re(λ)
-2
0
2
Im(λ)
(b) Antiperiodic case (m = 1)
Figure 5: Free-field spectrum of D
dw
with Wilson kernel in the standard construction
with different boundary conditions at N
s
= 8 (cf. Fig. 3)
imaginary axis via
π (λ) =
λ
1 −
1
%
λ
. (96)
We also point out the high degree of symmetry of the
spectrum in the case of Adams’ overlap. As noted before,
the effective Dirac operators in Boriçi’s and the optimal
construction converge towards Neuberger’s and Adams’
overlap operator for N
s
→ ∞, while in the standard con-
struction we find a modified overlap kernel.
B. U(1) gauge field case
While the free field is an interesting case, our main in-
terest is the performance of the Dirac operators in non-
11
0
0.5
1
1.5
2
Re(λ)
-1
-0.5
0
0.5
1
Im(λ)
std Bor
opt
(a) Wilson kernel
0
0.5
1
1.5
2
Re(λ)
-1
-0.5
0
0.5
1
Im(λ)
std Bor
opt
(b) Staggered Wilson kernel
Figure 6: Spectrum of %D
eff
at N
s
= 2 for the standard (std), Boriçi (Bor) and optimal (opt) construction.
0
0.5
1
1.5
2
Re(λ)
-1
-0.5
0
0.5
1
Im(λ)
std Bor
opt
(a) Wilson kernel
0
0.5
1
1.5
2
Re(λ)
-1
-0.5
0
0.5
1
Im(λ)
std Bor
opt
(b) Staggered Wilson kernel
Figure 7: Spectrum of %D
eff
at N
s
= 8 for the standard (std), Boriçi (Bor) and optimal (opt) construction.
trivial background fields. Dealing with the Schwinger
model, for the rest of the work we consider U(1) gauge
fields. We use quenched thermalized gauge fields follow-
ing the setup of Refs. [100, 101]. Note that while these
gauge fields originate from a quenched ensemble, there is
no problem in the Schwinger model to reweight them to
an arbitrary mass unquenched ensemble [100–103].
In this section, we start our investigation by focusing
on a few individual 20
2
configurations at an inverse cou-
pling of β = 5 to illustrate the qualitative features of the
spectra.
e. Kernel operators. In Fig. 9, we can see the kernel
spectra in a gauge background with Q = 1. As expected
in the Schwinger model, the branches stay much sharper
and well separated compared to the (3 + 1)-dimensional
QCD case [40–43].
f. Bulk operators. In Figs. 10 and 11, we show the
spectra of the bulk operators on the same gauge config-
uration as used in Fig. 9. Due to the use of a gauge
configuration with Q 6= 0, the effective operator is guar-
anteed to have |Q| exact zero modes in the limit N
s
→ ∞.
In this setting, we find N
s
· |Q| eigenvalues in the vicin-
12
0
0.5
1
1.5
2
Re(λ)
-1
-0.5
0
0.5
1
Im(λ)
DW Neub
(a) Wilson kernel
0
0.5
1
1.5
2
Re(λ)
-1
-0.5
0
0.5
1
Im(λ)
DW Adams
(b) Staggered Wilson kernel
Figure 8: Spectrum of %D
ov
with stereographic projection for domain wall (DW)
and standard (Neub/Adams) kernels.
0
0.5
1
1.5
2
2.5
3
3.5
4
Re(λ)
-1.5
-1
-0.5
0
0.5
1
1.5
Im(λ)
7.85×10
-2
2.23×10
0
9.31×10
0
8.73×10
-2
1.78×10
0
2.62×10
0
m
eff
∆
N
∆
GW
Wil
stag
Figure 9: Spectrum of kernel operators.
ity of the origin in the bulk spectrum, which are linked
to these zero modes. We can also see how the optimal
construction distorts the spectrum, effectively improving
chiral properties and resulting in a reduced m
eff
.
g. Effective operators. In Figs. 12 and 13, we plot
the spectra of the effective operators on the same gauge
field background as used in Fig. 9. We note that for
N
s
≥ 4 Boriçi’s construction outperforms the standard
construction with respect to all measures m
eff
, ∆
N
and
∆
GW
.
The optimal construction decreases most of these num-
bers even further. In Fig. 13b we see that m
eff
is already
comparable to the round-off error and, hence, we only
quote an upper bound. Note that as in a U(1) back-
ground field the kernel operator is in general not normal,
the inequality ||λ| − 1 | ≤ δ
max
does not have to be sat-
urated. Moreover it is evident that a smaller maximum
deviation δ
max
from the sign function on a given interval
does not necessarily translate to a smaller ∆
GW
, although
there is a strong correlation. For larger N
s
this problem
is cured by the fast convergence of optimal domain wall
fermions.
Let us remark that optimal domain wall fermions are
optimal in a very particular sense, namely the minimiza-
tion of δ
max
as defined in Eq. (95). As Ref. [66] suggests,
they are not optimal with respect to e.g. the number of
iterations needed for solving a linear system. In principle,
one could also formulate domain wall fermions optimized
with respect to other measures, such as the minimization
of ∆
GW
(which, however, might require more knowledge
about the spectrum).
Comparing domain wall fermions with a Wilson and
staggered Wilson kernel, we can see that in the case of
the standard construction m
eff
, ∆
N
and ∆
GW
are usually
of the same magnitude. However, for Boriçi’s and the
optimal construction a staggered Wilson kernel seems to
outperform the usual Wilson kernel in terms of chiral
symmetry violations in the U(1) background fields under
consideration.
For the rather artificial case N
s
= 2 we can make some
interesting observations. While for a staggered Wilson
kernel the relative performances of all formulations un-
der consideration is comparable, for a Wilson kernel the
standard formulation performs better than Boriçi’s and
markedly better than optimal.
13
-2 0 2 4
6
8
Re(λ)
-4
-2
0
2
4
Im(λ)
(a) Standard construction
-2 0 2 4
6
8
Re(λ)
-4
-2
0
2
4
Im(λ)
(b) Boriçi’s construction
-2 0 2 4
6
8
Re(λ)
-4
-2
0
2
4
Im(λ)
(c) Optimal construction
Figure 10: Spectrum of D
dw
with Wilson kernel for m = 0 at N
s
= 8.
-2 0 2 4
6
8
Re(λ)
-4
-2
0
2
4
Im(λ)
(a) Standard construction
-2 0 2 4
6
8
Re(λ)
-4
-2
0
2
4
Im(λ)
(b) Boriçi’s construction
-2 0 2 4
6
8
Re(λ)
-4
-2
0
2
4
Im(λ)
(c) Optimal construction
Figure 11: Spectrum of D
sdw
with staggered Wilson kernel for m = 0 at N
s
= 8.
h. Overlap operators. In Fig. 14, we show the cor-
responding overlap operators together with the stereo-
graphic projection of the spectrum. All of the quantities
m
eff
, ∆
N
and ∆
GW
vanish in exact arithmetic and are,
therefore, omitted in the figure labels.
C. Smearing
In realistic simulations, smearing is a commonly em-
ployed technique. As suggested in Ref. [42], it is expected
to be especially beneficial when employing a staggered
Wilson kernel (in 3 + 1 dimensions, this is more pro-
nounced). Hence, we consider three-step ape smeared
[104, 105] gauge field backgrounds with a smearing pa-
rameter α = 0.5, which is the maximal value within the
perturbatively reasonable range in two dimensions [106].
As a first test on the impact on the performance of
staggered domain wall fermions, we can find a direct
comparison at N
s
= 4 in Figs. 15 and 16 between an un-
smeared and a smeared version of a configuration both
for a Wilson and a staggered Wilson kernel. As we can
see, with three smearing steps, m
eff
, ∆
N
and ∆
GW
get
reduced significantly—in some cases of up to 2 orders
of magnitude—with the exception of m
eff
for standard
domain wall fermions with Wilson kernel.
D. Topology
As discussed earlier, for topological charges Q 6= 0, the
Atiyah-Singer index theorem guarantees the existence of
zero modes of the continuum Dirac operator. On the
lattice one can show the same for the overlap operators
defined in Eq. (80). As a consequence we observe ap-
proximate zero modes in the eigenvalue spectra of the
effective operators %D
eff
as illustrated in Fig. 17. As the
Vanishing Theorem holds in 1 + 1 dimensions, we find
these modes with a multiplicity of |Q|.
In addition, we can study topological aspects by em-
ploying the method in Ref. [97] to construct gauge config-
urations with given topological charge Q. These smooth
14
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 12: Spectrum of %D
eff
at N
s
= 2 for the standard (std), Boriçi (Bor) and optimal (opt) construction.
0
0.5
1
1.5
2
Re(λ)
-1
-0.5
0
0.5
1
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
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)
2.82×10
-4
3.26×10
-2
9.12×10
-2
3.55×10
-6
1.03×10
-3
1.01×10
-3
<3
×10
-8
5.03×10
-6
5.71×10
-6
m
eff
Δ
N
Δ
GW
std
Bor
opt
(b) Staggered Wilson kernel
Figure 13: Spectrum of %D
eff
at N
s
= 8 for the standard (std), Boriçi (Bor) and optimal (opt) construction.
configurations are fixed points with respect to the ape
smearing prescription. We construct these configurations
for a wide range of topological charges Q and evaluate the
measures m
eff
, ∆
N
and ∆
GW
for the effective operators.
15
0
0.5
1
1.5
2
Re(λ)
-1
-0.5
0
0.5
1
Im(λ)
DW Neub
(a) Wilson kernel
0
0.5
1
1.5
2
Re(λ)
-1
-0.5
0
0.5
1
Im(λ)
DW Adams
(b) Staggered Wilson kernel
Figure 14: Spectrum of %D
ov
with stereographic projection for domain wall (DW)
and standard (Neub/Adams) kernels.
0
0.5
1
1.5
2
Re(λ)
-1
-0.5
0
0.5
1
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
m
eff
∆
N
∆
GW
std
Bor
opt
(a) No smearing
0
0.5
1
1.5
2
Re(λ)
-1
-0.5
0
0.5
1
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
m
eff
∆
N
∆
GW
std
Bor
opt
(b) Three smearing iterations
Figure 15: Spectrum of %D
eff
with Wilson kernel and smearing parameter α = 0.5 at N
s
= 4
for the standard (std), Boriçi (Bor) and optimal (opt) construction.
In this setting, these measures are equal within numeri-
cal rounding errors for topological charge ±Q and, thus,
only depend on |Q|.
We find that m
eff
, ∆
N
and ∆
GW
are very small in
magnitude on these specific configurations compared to
thermalized configurations. Moreover, they increase with
larger values of |Q|. In Fig. 18, we can see two examples
of bulk spectra on a 20
2
lattice, which reveal a very clear
16
0
0.5
1
1.5
2
Re(λ)
-1
-0.5
0
0.5
1
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
m
eff
∆
N
∆
GW
std
Bor
opt
(a) No smearing
0
0.5
1
1.5
2
Re(λ)
-1
-0.5
0
0.5
1
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
m
eff
∆
N
∆
GW
std
Bor
opt
(b) Three smearing iterations
Figure 16: Spectrum of %D
eff
with staggered Wilson kernel and smearing parameter α = 0.5 at N
s
= 4
for the standard (std), Boriçi (Bor) and optimal (opt) construction.
(a) Configuration with Q = 0
(b) Configuration with Q = 1
(c) Configuration with Q = 3
Figure 17: Spectrum of %D
eff
with staggered Wilson kernel for various topological charges Q at N
s
= 4
for the standard (std), Boriçi (Bor) and optimal (opt) construction.
structure on these smooth background fields. E. Spectral flow
Another tool to investigate topological aspects on the
lattice, such as the index theorem, is the spectral flow
17
-2 0 2 4
6
8
Re(λ)
-4
-2
0
2
4
Im(λ)
(a) Wilson kernel
-2 0 2 4
6
8
Re(λ)
-4
-2
0
2
4
Im(λ)
(b) Staggered Wilson kernel
Figure 18: Spectrum of D
dw
in Boriçi’s construction at N
s
= 4 for a topological configuration with Q = 3.
[4, 107].
In the case of a Wilson kernel, one considers the eigen-
values {λ (m
f
)} of the Hermitian operator H
w
(m
f
) as
a function of m
f
. One can show that there is a one-
to-one correspondence between the eigenvalue crossings
of H
w
(m
f
) and the real eigenvalues of D
w
(m
f
). Fur-
thermore the low-lying real eigenvalues of D
w
(m
f
) corre-
spond to the would-be zero modes [97] and one can show
that for these modes the slope of the eigenvalue crossings
in the vicinity of m
f
= 0 equals minus the chirality. By
identifying the eigenvalues crossings occurring at small
values of m
f
as well as their slopes, one can then infer
the topological charge Q of the gauge field.
While originally the spectral flow was used for a Wilson
kernel, eventually the applicability to the staggered case
could be shown as well [31, 40, 41, 96, 108]. Spectral
flows of both the Wilson and staggered Wilson kernel
have been investigated in the literature before. Here we
want to illustrate the effectiveness of smearing.
In Figs. 19 and 20, we show the eigenvalue flow λ (m
f
)
of H
w
(m
f
) and H
sw
(m
f
) for the lowest 50 eigenvalues.
We consider a 12
2
gauge configuration at β = 1.8 with
topological charge Q = 2. We calculate the eigenvalue
flow on the unsmeared and three-step ape smeared con-
figuration with smearing parameter α = 0.5. For compar-
ison, we also show the corresponding topological gauge
configuration described in Sec. VI D. As one can see, the
use of smearing allows the unambiguous determination of
the topological charge Q, which on the rough configura-
tion without smearing is otherwise difficult. Finally, the
corresponding topological charge for Q = 2 is so smooth
that the two eigenvalue crossings close to the origin lie
on top of each other. This shows how beneficial the use
of smearing is when studying topological aspect using
spectral flows.
F. Approaching the continuum
In order to judge the performance of the different
fermion formulations when approaching the continuum,
we evaluated them on seven different ensembles with 1000
configurations each. We kept the physical volume con-
stant and considered the following lattices: 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. Together
with the smeared version of each configuration, we con-
sider N = 14 000 configurations in total.
We do not attempt to carry out a strict continuum
limit analysis, but we are interested in the relative perfor-
mance of the different formulations when the lattices be-
come finer. An indication for the performance are the chi-
ral symmetry violations on our finest lattice at β = 12.8.
We quote the median values in Table I. We restrict our-
selves to N
s
∈ {2, 4, 6} as for N
s
≥ 8 some values have
the same of order of magnitude as the rounding errors
and, thus, we are unable to quote precise numbers. We
observe that for large β the chiral symmetry violations
for the standard construction are comparable in the case
of a Wilson and a staggered Wilson kernel. For Boriçi’s
and the optimal construction, on the other hand, the vi-
olations are much lower for a staggered Wilson kernel,
often by 1 to 2 orders of magnitude. Note that for sim-
plicity we set the parameters λ
min
and λ
max
for optimal
domain wall fermions on a configuration basis. This is
intended to give an indication of the performance, while
in realistic simulations one would fix suitable values on
an ensemble basis after having projected out a number
of low-lying eigenmodes. One then has to find a compro-
mise between mapping small eigenvalues accurately and
keeping the overall approximation error small.
In general, we also observe that the optimal construc-
tion shows a better performance than Boriçi’s, while
18
−1
−0.5
0
0.5
1
−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1
λ(m)
Fermion mass m
f
(a) Unsmeared configuration
−1
−0.5
0
0.5
1
−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1
λ(m)
Fermion mass m
f
(b) Smeared configuration
−1
−0.5
0
0.5
1
−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1
λ(m)
Fermion mass m
f
(c) Topological configuration
Figure 19: Spectrum of the Wilson kernel for gauge fields with Q = 2.
−1
−0.5
0
0.5
1
−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1
λ(m)
Fermion mass m
f
(a) Unsmeared configuration
−1
−0.5
0
0.5
1
−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1
λ(m)
Fermion mass m
f
(b) Smeared configuration
−1
−0.5
0
0.5
1
−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1
λ(m)
Fermion mass m
f
(c) Topological configuration
Figure 20: Spectrum of the staggered Wilson kernel for gauge fields with Q = 2.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
CDF(λ
min
)
λ
min
β = 3.2
β = 5.0
β = 12.8
(a) Wilson kernel
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
CDF(λ
min
)
λ
min
β = 3.2
β = 5.0
β = 12.8
(b) Staggered Wilson kernel
Figure 21: Cumulative distribution function CDF for λ
min
of H
w
and H
sw
.
Boriçi’s construction performs better than the standard
construction. A interesting exception is m
eff
, where the
optimal construction with a Wilson kernel is not clearly
outperforming the other constructions. This is not unex-
pected as the optimal sign-function approximation has a
point of maximal deviation at λ
min
and as a result low-
lying eigenvalues are not mapped accurately on smooth
configurations. However, for larger N
s
this phenomenon
disappears and the optimal construction shows a superior
performance.
A related question is how the lowest eigenvalue λ
min
=
min
λ∈spec H
|λ| of the kernel operators H = H
w
and
H = H
sw
is distributed. Especially close to the origin
smooth approximations to the sign-function tend to be
inaccurate, so the mappings of small eigenvalues suffer
from large errors. In Fig. 21, we can find the numerically
determined cumulative distribution functions (CDFs) for
both kernel at three different β. As expected, for larger
β the tail of small values thins out and near-zero λ
min
be-
come infrequent. The CDFs for the Wilson and staggered
19
Kernel Construction N
s
m
eff
∆
N
∆
GW
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
0
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
0
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 I: Median values for the chiral symmetry violations on unsmeared 32
2
configurations at β = 12.8.
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 22: ∆
GW
of %D
eff
in Boriçi’s construction at N
s
= 4 as a function of 1/β.
Wilson kernel look very much alike and the probability to
encounter configurations where λ
min
≈ 0 is comparable.
Finally, an example of how chiral symmetry violations
vary with β can be found in Fig. 22. We show here the
violation of the Ginsparg-Wilson relation ∆
GW
of %D
eff
in Boriçi’s construction at N
s
= 4 as a function of 1/β.
We plot the median value together with the width of
the distribution, characterized by the q-quantile and the
(1 − q)-quantile, where q =
1 − erf
1/
√
2

/2 and erf
denotes the error function. With this choice 68.3 % of
the values are within the error bars. One can clearly see
how the effective operator with a staggered Wilson kernel
shows superior chiral properties when β is sufficiently
large. As expected, we observe that smearing improves
chiral properties in particular on the coarsest lattices.
20
VII. CONCLUSIONS
In this work, we gave an explicit construction of stag-
gered domain wall fermions and investigated some of
their basic properties in the free-field case, on quenched
thermalized gauge configurations in the Schwinger model
and on smooth topological configurations. It appears
that staggered domain wall fermions indeed work as ad-
vertised.
Moreover we could generalize existing modifications of
domain wall fermions, such as Boriçi’s and the optimal
construction to the staggered case. This gives rise to pre-
viously not considered truncated staggered domain wall
fermions and optimal staggered domain wall fermions.
These modified staggered domain wall fermions in par-
ticular show significantly smaller chiral symmetry viola-
tions than the traditional Wilson based formulations, at
least in our setting and with respect to the criteria used
in this work. These properties make formulations with
a staggered Wilson kernel potentially interesting when
studying phenomena, where chiral symmetry is of im-
portance.
It is not yet clear how our results in U(1) background
gauge fields will translate to QCD in 3 + 1 dimensions
with a SU(3) gauge group, but they are encouraging and
warrant further investigations.
ACKNOWLEDGMENTS
We thank Ting-Wai Chiu for helping us to validate
our implementation of optimal domain wall fermions and
Stephan Dürr for helpful comments on the manuscript.
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.
Parts of the computations were done on the computer
cluster of the University of Wuppertal.
Appendix A: Optimal weights
In the following, we give some example values for the
weight factors as defined in Sec. III C. To this end, let us
consider the free-field case in 1 + 1 dimensions with the
particular choices M
0
= 1 and N
s
= 8. For the Wilson
kernel we then have λ
min
= 1 and λ
max
= 3, for the
staggered Wilson kernel λ
min
= 1 and λ
max
=
√
2. The
corresponding example weights {ω
s
} for optimal domain
wall fermions can be found in Table II.
s ω
s
(λ
min
= 1, λ
max
= 3)
ω
s
λ
min
= 1, λ
max
=
√
2
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 II: ω
s
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