arXiv:2203.06116v2 [hep-lat] 8 Nov 2022
Locality of staggered overlap operators
Nuha Chreim
∗
, Christian Hoelbling
†
, and Christian Zielinski
‡
Bergische Universität Wuppertal, Gaußstraße 20, D-42119 Wuppertal, Germany
Abstract
We give an explicit proof for the locality of staggered overlap operator s. The proof covers
the original two flavor construction by Adams as well as a single flavor version. As in the
case of Neuberger’s operator, an admissibility condition for the gaug e fields is required.
1 Introduction and motivation
As Adams has shown [1], it is possible to construct chirally symmetric lattice fermions based on
the staggered discretization. While Adams’ original construction provided a two flavor operator,
single flavor versions were found soon after [2, 3]. These staggered chiral fermions are obtained
by first adding a mass term [4] to the s taggered operator, followed by an overlap construction
[5], which contains an inverse square root. It is thus evident that staggered chiral fermions
are not ultralocal by construction and their locality needs to be proven. Numerically, Ref. [6]
found strong evidence in support of the locality of Adams’ original two flavor operator. In the
free case, one can furthermore show that the lifting of the doubler modes is achieved via flavor
dependent mass term [1, 2, 7]. In addition, the index theorem has been established for the two
flavor operator [8] and the correct continuum limit of the index was found in [9]. In this paper,
we give an analytic proof for the locality of staggered overlap fermions, for both the single and
two flavor cases. The general strategy we employ is quite similar to the one used by Hernández,
Jansen and Lüscher to demonstrate the locality of the original Neuberger operator [10]. We
will start in sec. 2 by expanding the inverse square root as a series of Legendre polynomials,
which can be shown to be lo cal if a spectral condition of the kernel operator is fulfilled. This
spectral condition involves an upper as well as a lower bound on the kernel operator. In sec. 3
we will show that both bounds are fulfilled for Adams’ original two flavor construction, provided
an admissibility condition of the form k
1
−P k < ε is fulfilled by all plaquettes P of the gauge
field. The exact value of ε will depend on the details of the action, specifically the neg ative mass
parameter s and the Wilson parameter r. We then turn to a single flavor staggered operator and
show that similar bounds also hold in this case.
2 Locality
2.1 Staggered overlap Dirac operator
Let us first introduce the staggered overlap Dirac op erator
D
so
=
1
a
1
+A/
√
A
†
A
(2.1)
∗
nuha.chreim@uni-wuppertal.de
†
hch@uni-wuppertal.de
‡
email@czielinski.de
1
2.1 Staggered overlap Dirac operator 2 LOCALITY
with
A = aD
sw
− rs
1
D
sw
= D
st
+ W
st
(2.2)
where r is the Wilson parameter and 0 < s < 2 is the negative mass term of the kernel operator.
The staggered operator is defined as
D
st
= η
µ
∇
µ
(2.3)
with
(η
µ
)
x
= (−1)
P
ν<µ
x
ν
(2.4)
and the symmetric derivative operator
∇
µ
=
1
2a
(T
µ+
− T
µ−
) . (2.5)
The T
µ±
are parallel transports defined as
(T
µ+
)
xy
= U
µ
(x)δ
x+ˆµ,y
(T
µ−
)
xy
= U
†
µ
(y)δ
x−ˆµ,y
. (2.6)
The staggered Wilson term W
st
reads
W
st
=
r
a
1
−M
(2)
(2.7)
in the two flavor case [1, 8, 11] and
W
st
=
r
a
2 ·
1
+M
(1)
(2.8)
in the one flavor case [2, 3]. The operators M
(N
f
)
are in turn given
1
by
M
(2)
= ǫη
5
C M
(1)
= i η
12
C
12
+ i η
34
C
34
(2.9)
with the phase factors
η
5
= η
1
η
2
η
3
η
4
(2.10)
ǫ
x
= (−1)
P
ν
x
ν
(2.11)
(η
µν
)
x
= −η
νµ
= (−1)
P
ν
ρ=µ+1
x
ρ
for µ ≤ ν (2.12)
and the diagonal hopping terms
C = (C
1
C
2
C
3
C
4
)
sym
=
1
4!
P
αβγδ
C
α
C
β
C
γ
C
δ
(2.13)
C
µν
=
1
2
{C
µ
, C
ν
} (2.14)
where
C
µ
=
1
2
(T
µ+
+ T
µ−
) (2.15)
and P
αβγδ
denotes the permutation symbol
P
αβγδ
=
1 α, β, γ, δ is a permutation of 1, 2, 3, 4
0 else.
(2.16)
The kernel A is ultralocal, but due to the
A
†
A
−1/2
term the staggered overlap Dirac operator
D
so
is not. However, if the matrix elements (D
so
)
x,y
of the staggered overlap operator are
decaying exponentially for large distances kx − yk with a decay constant ∝ a
−1
, then we recover
a local operator in the continuum limit.
1
Note that in principle more general single flavor terms are allowed [3]. These are, however, not substantially
different and the generalization is straightforward.
2
2.2 Legendre series expansion 2 LOCALITY
2.2 Legendre series expansion
Following the strategy employed in Ref. [10] we begin by expanding
A
†
A
−1/2
in a series of
Legendre polynomials. In order to make the expansion convergent we impose the following
inequality, which we will show in sect. 3:
0 < u ≤ A
†
A ≤ v < ∞. (2.17)
The inequality stands for the corresponding inequality between the expectation values of the
operators in arbitrary normalizable states. We also explicitly assume that u < v. In the following
we can s et u = λ
min
and v = λ
max
as noted in Ref. [12].
The Legendre polynomials P
k
(z) can be defined through the ex pansion of the generating function
1 − 2tz + t
2
−1/2
=
∞
X
k=0
t
k
P
k
(z) . (2.18)
We can now set
z =
(λ
min
+ λ
max
)
1
−2A
†
A
λ
max
−λ
min
(2.19)
and due to eq. (2.17) find that this operator has norm kzk = 1. Here and in the following
k·k = k·k
2
≡ σ
max
(·) refers to the spectral norm and σ
max
refers to the largest singular value.
Then the property |P
k
(x)| ≤ 1 ∀x ∈ [−1, 1] together with kzk = 1 translates to
kP
k
(z)k ≤ 1. (2.20)
It follows that eq. (2.18) is norm convergent for our choice of z for all t satisfying |t| < 1. Due to
eq. (2.17), we can now introduce θ through
cosh θ =
λ
max
+ λ
min
λ
max
− λ
min
, θ > 0, (2.21)
and set
t = e
−θ
, (2.22)
which implies 0 < t ≤ 1, so that the series is convergent. Note that this allows us to express t as
t = cosh θ −
p
cosh
2
θ − 1 =
√
λ
max
−
√
λ
min
√
λ
max
+
√
λ
min
. (2.23)
From eq. (2.18) we thus obtain
1 − 2tz + t
2
−1/2
=
1 −
2t
λ
max
− λ
min
λ
min
+ λ
max
− 2A
†
A
+ t
2
−1/2
=
1 − 2e
−θ
cosh θ +
4t
λ
max
− λ
min
A
†
A + e
−2θ
−1/2
=
r
λ
max
− λ
min
4t
A
†
A
−1/2
=
√
λ
max
+
√
λ
min
2
A
†
A
−1/2
(2.24)
and therefore
A
†
A
−1/2
= κ
∞
X
k=0
t
k
P
k
(z) (2.25)
with κ = 2/
√
λ
max
+
√
λ
min
.
3
2.3 Locality of the inverse square root 2 LOCALITY
2.3 Locality of the inverse square root
The lack of ultralocality stems from the
A
†
A
−1/2
term, hence it is sufficient to establish the
locality of that term in the sense defined earlier. We start by defining the kernel G (x, y) via
G (x, y) =
(A
†
A)
−1/2
xy
. (2.26)
Similarly, we define the kernels of the P
k
(z) v ia
G
k
(x, y) = (P
k
(z))
xy
(2.27)
and use eq. (2.25) to obtain
G (x, y) = κ
∞
X
k=0
t
k
G
k
(x, y) . (2.28)
The norm convergence of the Legendre expansion implies the absolute convergence of this series
for all x and y. From eq. (2.20) and eq. (2.27) we infer that
kG
k
(x, y)k ≤ 1, ∀x ∀y ∀ k, (2.29)
where the norm is in color space.
Because P
k
(z) is a polynomial in A
†
A and A is an ultralocal operator, we find that G
k
(x, y)
vanishes unless x and y are sufficiently close to each other. If we introduce the Manhattan
distance k·k
1
, we have
G
k
(x, y) = 0, ∀k <
1
2ℓa
kx − yk
1
, (2.30)
where ℓ is the range of the operator A in lattice units, i.e., the maximum Manhattan distance in
lattice units between points coupled by the operator. For two flavor staggered Wilson fermions
we have ℓ = 4, for one flavor staggered Wilson fermions ℓ = 2 and for Wilson fermions ℓ = 1.
Using the shorthand notation d = kx − yk
1
/(2ℓa) we find
kG (x, y)k = κ
∞
X
k=d
t
k
kG
k
(x, y)k
≤ κ
∞
X
k=d
t
k
=
κ
1 − t
t
d
=
κ
1 − t
exp
−
θ
2ℓa
kx − yk
1
=
1
√
λ
min
exp
−
1
ξ
kx − yk
1
(2.31)
and thus an exponential falloff with the decay constant
2
ξ
−1
=
θ
2ℓa
=
1
2ℓa
log
√
λ
max
+
√
λ
min
√
λ
max
−
√
λ
min
∝
1
a
. (2.32)
This establishes the locality of
A
†
A
−1/2
providing eq. (2.17) holds with the spectral bounds
given by u = λ
min
and v = λ
max
. The equivalent of this particular form for usual overlap
fermions was derived in Ref. [12].
2
Note that the log term may provide subleading corrections to this behavior only.
4
3 BOUNDS ON A
†
A
Let us finally remark that this result can be slightly generalised in the case of a single isolated
zero or near zero mode λ
min
. As shown in sect. 2.4 of Ref. [10], one can treat a s ingle isolated
zero or near zero mode separately and still establish locality. In that case we identify the lower
spectral bound u = λ
2
with the second smallest eigenvalue of A
†
A. If λ
min
< u/2, locality can
again be established [10].
3 Bounds on A
†
A
We now need to establish the spectral bounds as defined in eq. (2.17) for the kernel operator. We
first derive some useful identities and then establish the upper bound, which is straightforward.
The main task is then to establish the lower bound, which we do separately for the two and one
flavor case. In both instances, the bound can b e established given an admissibility condition for
the gauge fields.
3.1 Some useful identities
We first note that the parallel transports fulfill the relations T
µ−
= T
†
µ+
= T
−1
µ+
, which implies
that the T
µ±
are unitary and thus have singular values 1, i.e. kT
µ±
k = 1. The covariant second
derivative operator is given by
∆
µ
= T
µ+
+ T
µ−
−2, (3.1)
so we can recast eq. (2.15) as
C
µ
= 1 +
∆
µ
2
. (3.2)
Using this relation we find
C
2
µ
= 1 +
1
4
(T
2
µ+
+ T
2
µ−
− 2). (3.3)
Defining
V
µ
=
1
4
(T
2
µ+
+ T
2
µ−
−2) (3.4)
it follows that
C
2
µ
= 1 + V
µ
. (3.5)
From eq. (2.15) we also find
kC
µ
k ≤
1
2
(kT
µ+
k + kT
µ−
k) = 1, (3.6)
which implies
kM
(2)
k = kη
5
Cǫk ≤ 1 (3.7)
and, since both η
5
and ǫ commute with C and square to the identity, M
(2)
2
= C
2
. Another
useful identity is
a
2
∇
2
µ
= V
µ
(3.8)
which, together with the anti Hermiticity condition ∇
†
µ
= −∇
µ
, implies that
0 ≤ a
2
∇
†
µ
∇
µ
= −V
µ
. (3.9)
Additionally, the Hermiticity condition C
†
µ
= C
µ
implies that C
2
µ
≥ 0 and thus 1 + V
µ
≥ 0.
Next, we want to find a more explicit expressions for A
†
A. Noting that
∇
µ
η
ν
=
(
η
ν
∇
µ
µ ≥ ν,
−η
ν
∇
µ
µ < ν,
(3.10)
5
3.2 Upper bound 3 BOUNDS ON A
†
A
we find
X
µ,ν
η
µ
∇
µ
η
ν
∇
ν
= ∇
2
+
X
µ>ν
η
µ
η
ν
[∇
µ
, ∇
ν
] , (3.11)
where we have introduced the shorthand notation
∇
2
=
X
µ
∇
µ
∇
µ
. (3.12)
We then find
A
†
2
A
2
= −a
2
∇
2
− a
2
X
µ>ν
η
µ
η
ν
[∇
µ
, ∇
ν
] + r
2
1
(1 − s) − M
(2)
2
− ar
h
M
(2)
, η
µ
∇
µ
i
(3.13)
in the two flavor case and
A
†
1
A
1
= −a
2
∇
2
− a
2
X
µ>ν
η
µ
η
ν
[∇
µ
, ∇
ν
] + r
2
1
(2 − s) + M
(1)
2
+ ar
h
M
(1)
, η
µ
∇
µ
i
(3.14)
in the one flavor case.
3.2 Upper bound
Using kT
µ±
k = 1 we find the following bounds
ka∇
µ
k ≤
1
2
(kT
µ+
k + kT
µ−
k) ≤ 1, (3.15)
kaη
µ
∇
µ
k ≤ 4, (3.16)
kC
µ
k =
1
2
kT
µ+
+ T
µ−
k ≤ 1, (3.17)
kCk =
(C
1
C
2
C
3
C
4
)
sym
≤
1
4!
· 4! ·
Y
µ
kC
µ
k ≤ 1, (3.18)
and using eq . (3.7) we find
r
1
(1 − s) − M
(2)
≤ |r|(2 − s). (3.19)
Putting all this together, we find
kA
2
k =
w
w
w
aη
µ
∇
µ
+ r
1
(1 − s) − M
(2)
w
w
w
≤ 4 + |r|(2 − s). (3.20)
The same bound holds for A
†
2
and so
kA
†
2
A
2
k ≤ kA
†
2
kkA
2
k ≤ (4 + |r|(2 − s))
2
(3.21)
is uniformly bounded from above for all r and s and we can establish the existence of v in
eq. (2.17) in the two flavor case.
For the one flavor case we note that
kC
µν
k =
1
2
k{C
µ
, C
ν
}k ≤ 1, (3.22)
from which it follows that
kM
(1)
k ≤ kC
12
k + kC
34
k ≤ 2. (3.23)
6
3.3 Lower bound 3 BOUNDS ON A
†
A
Hence we find
w
w
w
1
(2 − s) + M
(1)
w
w
w
≤ 4 − s (3.24)
and it follows, similarly to the two flavor case, that
kA
1
k ≤ 4 + |r|(4 − s). (3.25)
Since A
†
1
does obey the same bound, we obtain
kA
†
1
A
1
k ≤ (4 + |r|(4 − s))
2
. (3.26)
This establishes the existence of v in eq. (2.17) in the single flavor case as well.
3.3 Lower bound
As A
†
A is Hermitian and positive semidefinite we are left with showing the absence of zero-
modes. H owever, in general this operator can have zero-modes for certain gauge configurations,
therefore no uniform positive lower bound exists. Zero-modes can only be excluded if we assume
the gauge field to be sufficiently smooth. In our case let us assume that
k
1
−P k < ε for all plaquettes P. (3.27)
As a consequence of the smoothness condition, we obtain the following relations (see app. A)
a
2
[∇
µ
, ∇
ν
]
< ε, k[C
µ
, C
ν
]k < ε, ka[C
µ
, ∇
ν
]k < ε. (3.28)
3.3.1 Lower bound on the two flavor operator A
†
2
A
2
There are four terms in
A
†
2
A
2
= −a
2
∇
2
−
X
µ>ν
η
µ
η
ν
a
2
[∇
µ
, ∇
ν
] + r
2
1
(1 − s) − M
(2)
2
− ar
h
M
(2)
, η
µ
∇
µ
i
, (3.29)
for which we will find bounds individually. We will consider t he case 0 < r ≤ 1 first and derive
a bound for r > 1 later
3
.
The first and third term
We first look at −a
2
∇
2
+ r
2
C
2
, where M
(2)
2
= C
2
is used. Using inequality (3.28) we find (cf.
app. A)
kC
2
− (C
2
1
C
2
2
C
2
3
C
2
4
)
sym
k < 9ε. (3.30)
3
The r < 0 case can be covered by the simple replacement of r → |r| in the b ounds. However, negative r do
not represent a physically different system compared t o positive r and will therefore not be considered further.
7
3.3 Lower bound 3 BOUNDS ON A
†
A
Using eqs. (3.5), (3.8) and (3.9), we furthermore see that for 0 < r ≤ 1
−a
2
∇
2
+ r
2
C
2
> − a
2
∇
2
+ r
2
(C
2
1
C
2
2
C
2
3
C
2
4
)
sym
− 9r
2
ε
= −
X
µ
V
µ
+ r
2
1
4!
P
αβγδ
(1 + V
α
)(1 + V
β
)(1 + V
γ
)(1 + V
δ
) − 9r
2
ε
= −
X
µ
V
µ
+ r
2
+ r
2
X
µ
V
µ
+
1
2
r
2
X
µ6=ν
V
µ
V
ν
+
1
3!
r
2
X
µ6=ν6=α6=µ
V
µ
V
ν
V
α
+ r
2
(V
1
V
2
V
3
V
4
)
sym
− 9r
2
ε
=r
2
− (1 − r
2
)
X
µ
V
µ
+
1
2
r
2
X
µ6=ν
V
µ
V
ν
+
1
3!
r
2
X
µ6=ν6=α6=µ
V
µ
V
ν
V
α
+ r
2
(V
1
V
2
V
3
V
4
)
sym
− 9r
2
ε
≥r
2
1 +
1
2
X
µ6=ν
V
µ
V
ν
+
1
3!
X
µ6=ν6=α6=µ
V
µ
V
ν
V
α
+ (V
1
V
2
V
3
V
4
)
sym
− 9ε
. (3.31)
Using the relation (3.9), we conclude that
V
µ
V
ν
= (−V
µ
)(−V
ν
) ≥ 0, (3.32)
so that each contribution to the two-product term as well as the four-product term is positive
semidefinite. We use these properties and 1 + V
µ
≥ 0 to obtain
−a
2
∇
2
+ r
2
C
2
> r
2
1 +
1
2
X
µ6=ν
V
µ
V
ν
+
1
3!
X
µ6=ν6=α6=µ
V
µ
V
ν
V
α
− 9ε
> r
2
1 +
1
3!
X
µ6=ν6=α6=µ
V
µ
V
ν
(V
α
+ 1) − 9ε
≥ r
2
(1 − 9ε) . (3.33)
Using eq. (3.7), we finally obtain
−a
2
∇
2
+ r
2
1
(1 − s) − M
(2)
2
= −a
2
∇
2
+ r
2
C
2
− 2r
2
(1 − s)M
(2)
+ r
2
(1 − s)
2
1
≥ r
2
1 − 9ε −2|1 − s | + |1 − s|
2
= r
2
(1 − |1 − s|)
2
− 9r
2
ε (3.34)
for 0 < r ≤ 1. For the case r > 1 we can decompose
−a
2
∇
2
+ r
2
1
(1 − s) −M
(2)
2
= −a
2
∇
2
+
1
(1 − s) − M
(2)
2
+ (r
2
− 1)
1
(1 − s) − M
(2)
2
(3.35)
and, since r
2
− 1 > 0, observe that the last term is positive semidefinite. The first two terms,
however, just correspond to the r = 1 case, so the r = 1 lower bound also applies for the r > 1
case. All together we thus have
− a
2
∇
2
+ r
2
1
(1 − s) − M
(2)
2
>
r
2
(1 − |1 − s|)
2
− 9r
2
ε 0 < r ≤ 1,
(1 − |1 − s|)
2
−9ε r > 1.
(3.36)
8
3.3 Lower bound 3 BOUNDS ON A
†
A
The second term
As a result of eq. (3.28) we find
X
µ>ν
η
µ
η
ν
a
2
[∇
µ
, ∇
ν
]
≤
X
µ>ν
a
2
[∇
µ
, ∇
ν
]
< 6ε, (3.37)
so that we obtain the lower bound
−
X
µ>ν
η
µ
η
ν
a
2
[∇
µ
, ∇
ν
] > −6ε (3.38)
for the second term.
The fourth term
From the commutation properties
C
µ
η
ν
=
(
η
ν
C
µ
µ ≥ ν,
−η
ν
C
µ
µ < ν,
(3.39)
it f ollows that Cη
µ
= (−1)
µ+1
η
µ
C. Similarly one can show that ∇
µ
η
5
= (−1)
µ
η
5
∇
µ
. Using
these relations we find
h
M
(2)
, η
µ
∇
µ
i
= (ǫη
5
Cη
µ
∇
µ
− η
µ
∇
µ
ǫη
5
C)
= ǫ (η
5
Cη
µ
∇
µ
+ η
µ
∇
µ
η
5
C)
= ǫ
η
5
η
µ
(−1)
µ+1
C∇
µ
+ η
5
η
µ
(−1)
µ
∇
µ
C
= ǫη
5
η
µ
(−1)
µ+1
[C, ∇
µ
] . (3.40)
From eqs. (3.6) and (3.28) we can then conclude that
a
h
M
(2)
, η
µ
∇
µ
i
≤ a
X
µ
k[C, ∇
µ
]k
≤ a
X
µν
k[C
ν
, ∇
µ
]k
<
X
µ6=ν
ε
= 12ε (3.41)
and thus we obtain the lower bound
ar
h
M
(2)
, η
µ
∇
µ
i
> −12rε (3.42)
for all r > 0.
Final lower bound
Combining eqs. 3.36),(3.38) and (3.42), we get a lower bound for the two flavor operator
A
†
2
A
2
>
r
2
(1 − |1 − s|)
2
−(6 + 12r + 9r
2
)ε 0 < r ≤ 1,
(1 − |1 − s|)
2
− (15 + 12r)ε r > 1.
(3.43)
9
3.3 Lower bound 3 BOUNDS ON A
†
A
3.3.2 Lower bound on the one flavor operator A
†
1
A
1
We will now try to find a lower bound on the operator
A
†
1
A
1
= −a
2
∇
2
−
X
µ>ν
η
µ
η
ν
a
2
[∇
µ
, ∇
ν
] + r
2
2 ·
1
+M
(1)
− s
1
2
+ ar
h
M
(1)
, η
µ
∇
µ
i
, (3.44)
by finding a bound of each term separately. Since the second term is the same as in the two
flavor case, we can take the previous result eq. (3.38). Once again, we consider the case 0 < r ≤ 1
first.
The first and third terms
We start by observing that
C
2
µν
=
1
4
(C
µ
C
ν
+ C
ν
C
µ
)
2
=
1
4
(C
µ
C
ν
C
µ
C
ν
+ C
µ
C
ν
C
ν
C
µ
+ C
ν
C
µ
C
µ
C
ν
+ C
ν
C
µ
C
ν
C
µ
)
>
1
4
(C
2
µ
C
2
ν
− ε + C
2
µ
C
2
ν
− 2ε + C
2
ν
C
2
µ
− 2ε + C
2
ν
C
2
µ
− ε)
=
C
2
µ
C
2
ν
+ C
2
ν
C
2
µ
− 3ε
2
, (3.45)
where we have used eq. (3.28). For 0 < r ≤ 1 we thus obtain the bound
−a
2
∇
2
+ r
2
(
1
+ M
(1)
)
2
= −a
2
∇
2
+ r
2
(1 + iη
12
C
12
+ iη
34
C
34
)
2
= −
X
µ
V
µ
+ r
2
(C
2
12
+ C
2
34
+ {(1 + iη
12
C
12
), (1 + iη
34
C
34
)} − 1)
> −
X
µ
V
µ
+ r
2
C
2
1
C
2
2
+ C
2
2
C
2
1
− 3ε
2
+
C
2
3
C
2
4
+ C
2
4
C
2
3
− 3ε
2
− 1
= −
X
µ
V
µ
+
r
2
2
({(1 + V
1
), (1 + V
2
)} + {(1 + V
3
), (1 + V
4
)} − 2 − 6ε)
= −
X
µ
V
µ
+
r
2
2
2 + 2
X
µ
V
µ
+ {V
1
, V
2
} + {V
3
, V
4
} − 6ε
!
≥ −(1 − r
2
)
X
µ
V
µ
+ r
2
(1 − 6ε)
≥ r
2
− 6r
2
ε. (3.46)
For the general case of 0 < s < 2 we use
1
+M
(1)
≥ −1, (3.47)
which follows from
M
(1)
≤ 2, to find
−a
2
∇
2
+ r
2
(2 − s)
1
+ M
(1)
2
= −a
2
∇
2
+ r
2
(1 − s)
1
+
1
+ M
(1)
2
= −a
2
∇
2
+ r
2
1
+ M
(1)
2
+ r
2
(1 − s)
2
1
+ 2r
2
(1 − s)
1
+ M
(1)
> r
2
− 6r
2
ε + r
2
|1 − s|
2
− 2r
2
|1 − s |
= r
2
(1 − |1 − s|)
2
− 6r
2
ε. (3.48)
10
4 CONCLUSION
The lower bound of the first and third term for 0 < r ≤ 1 is thus given by
− a
2
∇
2
+ r
2
(2 − s)
1
+M
(1)
2
> r
2
(1 − |1 − s|)
2
− 6r
2
ε. (3.49)
For the r > 1 case we can again show that the r = 1 bound holds with the same argument used
in eq. (3.35). We thus obtain the general lower bound
− a
2
∇
2
+ r
2
(2 − s)
1
+M
(1)
2
>
r
2
(1 − |1 − s|)
2
− 6r
2
ε 0 < r ≤ 1,
(1 − |1 − s|)
2
− 6ε r > 1.
(3.50)
The fourth term
Let us first decompose the mass term
a[M
(1)
, η
µ
∇
µ
] = a i([η
12
C
12
, η
µ
∇
µ
] + [η
34
C
34
, η
µ
∇
µ
]) (3.51)
and look at the first of the two commutators. We have
a i[η
12
C
12
, η
µ
∇
µ
] = a i(η
12
[C
12
, η
µ
]∇
2
+ η
µ
η
12
[C
12
, ∇
µ
] + η
µ
[η
12
, ∇
µ
]C
12
)
= a i(−2η
12
η
2
C
12
∇
2
+ η
µ
η
12
[C
12
, ∇
µ
] + 2η
2
η
12
∇
2
C
12
)
= a i(−1)
δ
µ,2
η
µ
η
12
[C
12
, ∇
µ
], (3.52)
which results in
ka i[η
12
C
12
, η
µ
∇
µ
]k = ka i(−1)
δ
µ,2
η
µ
η
12
[C
12
, ∇
µ
]k
≤
a
2
(k[C
1
C
2
, ∇
µ
]k + k[C
2
C
1
, ∇
µ
]k)
≤
a
2
(kC
1
[C
2
, ∇
µ
]k + k[C
1
, ∇
µ
]C
2
k + kC
2
[C
1
, ∇
µ
]k+ k[C
2
, ∇
µ
]C
1
k). (3.53)
With eqs. (3.28) and (3.17) we thus obtain the upper bound
ka i[η
12
C
12
, η
µ
∇
µ
]k < 2ε (3.54)
for the first term. Similarly, we obtain for the second term
ka i[η
34
C
34
, η
µ
∇
µ
]k = ka i(−1)
δ
µ,4
η
µ
η
34
[C
34
, ∇
µ
]k < 2ε (3.55)
and thus conclude
ar[M
(1)
, η
µ
∇
µ
] > −4rε. (3.56)
Final lower bound
Combining eqs. (3.50), (3.38) and (3.56), we get a lower bound for the single flavor operator
A
†
1
A
1
>
r
2
(1 − |1 − s|)
2
− (6 + 4r + 6r
2
)ε 0 < r ≤ 1
(1 − |1 − s|)
2
− (12 + 4r)ε r > 1
(3.57)
4 Conclusion
In this note we have proven that, when the admissibility condition k
1
−P k < ε is imposed on
every plaquette P , both one and two flavor staggered overlap operators are local. In particular,
we can perform a Legendre expansion of the inverse square root of A
†
A, which is convergent if
11
4 CONCLUSION
the spectral condition of eq. (2.17) is fulfilled. From eqs. (3.43) and (3.57), we find that this is
the case when
ε <
r
2
(1 − |1 − s|)
2
6 + 12r + 9r
2
two flavor, 0 < r ≤ 1, (4.1)
ε <
(1 − |1 − s|)
2
15 + 12r
two flavor, r > 1, (4.2)
ε <
r
2
(1 − |1 − s|)
2
6 + 4r + 6r
2
single flavor, 0 < r ≤ 1, (4.3)
ε <
(1 − |1 − s|)
2
12 + 4r
single flavor, r > 1, (4.4)
which is dependent on the projection point s and the Wilson parameter r. The staggered overlap
operator is thus conceptually on the same f ooting as the standard overlap operator with a Wilson
kernel.
12
A PLAQUETTE DEPENDENT COMMUTATORS
Appendix A Plaquette dependent commutators
A.1 Representations of the plaquette
Since it is essential for the proof to have a bound on the plaqu ette, we first want to show how
the plaquette can be represented. Let us define the plaquette as the operator
(P
µν
)
xy
= U
µ
(x)U
ν
(x + ˆµ)U
†
µ
(x + ˆν)U
†
ν
(x)δ
x,y
. (A.1)
We find that
(T
µ+
T
ν+
T
µ−
T
ν−
)
xy
= U
µ
(x)δ
x+ˆµ,z
U
ν
(z)δ
z+ˆν,t
U
†
µ
(u)δ
t−ˆµ,u
U
†
ν
(y)δ
u−ˆν,y
= U
µ
(x)U
ν
(x + ˆµ)U
†
µ
(x + ˆν)U
†
ν
(y)δ
x,y
= (P
µν
)
xy
(A.2)
or equivalently
P
µν
= T
µ+
T
ν+
T
µ−
T
ν−
. (A.3)
Similarly, we can define plaquettes into negative coordinate directions as
P
(−µ)ν
= T
µ−
T
ν+
T
µ+
T
ν−
, (A.4)
P
µ(−ν)
= T
µ+
T
ν−
T
µ−
T
ν+
, (A.5)
P
(−µ)(−ν)
= T
µ−
T
ν−
T
µ+
T
ν+
. (A.6)
With these, we can find commutation relations among the T
µ±
(µ 6= ν) as
[T
µ+
, T
ν+
] = T
µ+
T
ν+
− T
ν+
T
µ+
= T
µ+
T
ν+
(1 − T
ν−
T
µ−
T
ν+
T
µ+
)
= T
µ+
T
ν+
(1 − P
(−ν)(−µ)
) (A.7)
and similarly for other combinations.
A.2 Implications for some commutators
We will need the commutator
a
2
[∇
µ
, ∇
ν
] =
1
4
([T
µ+
, T
ν+
] + [T
µ−
, T
ν−
] − [T
µ+
, T
ν−
] − [T
µ−
, T
ν+
])
=
1
4
(T
µ+
T
ν+
(1 − P
(−ν)(−µ)
) + T
µ−
T
ν−
(1 − P
νµ
)
− T
µ+
T
ν−
(1 − P
ν(−µ)
) − T
µ−
T
ν+
(1 − P
(−ν)µ
)), (A.8)
where we used eq . (2.5). Imp os ing a smoothness condition
k
1
−(P
µν
)
xx
k < ε (A.9)
on every plaquette and remembering that all kT
µ±
k = 1, we find that
a
2
k[∇
µ
, ∇
ν
]k <
ε
4
(kT
µ+
T
ν+
k + kT
µ−
T
ν−
k + kT
µ+
T
ν−
k + kT
µ−
T
ν+
k)
= ε. (A.10)
13
A.2 Implications for some commutators A PLAQUETTE DEPENDENT COMMUTATORS
Similarly we find
[C
µ
, C
ν
] =
1
4
([T
µ+
, T
ν+
] + [T
µ−
, T
ν+
] + [T
µ+
, T
ν−
] + [T
µ−
, T
ν−
]) (A.11)
=
1
4
(T
µ+
T
ν+
(1 − P
(−ν)(−µ)
) + T
µ−
T
ν+
(1 − P
(−ν)µ
)
+ T
µ+
T
ν−
(1 − P
ν(−µ)
) + T
µ−
T
ν−
(1 − P
νµ
)) (A.12)
and thus
k[C
µ
, C
ν
]k < ε. (A.13)
Using the fact that kC
µ
k ≤ 1, we can also infer that
[C
µ
, C
ν
]
n
Y
i=1
C
α
i
< ε (A.14)
for any number n of additional C
α
terms. We thus see that
kC
2
− (C
2
1
C
2
2
C
2
3
C
2
4
)
sym
k < N ε, (A.15)
where N is determined by the number of commutations we have to perform to bring the terms
in C
2
into the correct order. Let us first rewrite
C
2
− (C
2
1
C
2
2
C
2
3
C
2
4
)
sym
=
1
4!
P
αβγδ
(C
α
C
β
C
γ
C
δ
C − C
2
α
C
2
β
C
2
γ
C
2
δ
). (A.16)
For each term in the symmetrization bracket we now perform the commutations in two steps.
First we bring the terms in C into order, so we are left with (C
α
C
β
C
γ
C
δ
)
2
. For each of the 4!
products in C this requires a different number of commutations, namely
Number of commutations : 0 1 2 3 4 5 6
Number of products : 1 3 5 6 5 3 1
On average we thus have 3 commutations in this first step. From there </