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 </