arXiv:1312.7230v1 [hep-lat] 27 Dec 2013

Continuum limit of the axial anomaly and index for

the staggered overlap Dirac operator: An overview

David H. Adams

Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371

E-mail: dhadams@ntu.edu.sg

Reetabrata Har

∗

Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371

E-mail: REET0002@e.ntu.edu.sg

Yiyang Jia

Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371

E-mail: yyjiahbar@gmail.com

Christian Zielinski

Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371

E-mail: zielinski@pmail.ntu.edu.sg

Evaluation of the continuum limit of the axial anomaly and index is sketched for the staggered

overlap Dirac operator. There are new complications compared to the usual overlap case due

to the distribution of the spin and ﬂavor components around lattice hypercubes in the staggered

formalism. The index is found to correctly reproduce the continuum index, but for the axial

anomaly this is only true after averaging over the sites of a lattice hypercube.

31st International Symposium on Lattice Field Theory - LATTICE 2013

July 29 - August 3, 2013

Mainz, Germany

∗

Speaker.

c

Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/

Axial anomaly and index for the staggered overlap Dirac operator Reetabrata Har

1. Introduction

Overlap fermions, as a massless lattice fermion formulation, have exact zero-modes with deﬁ-

nite chirality and hence a well-deﬁned index [1]. Furthermore, they satisfy [1] the Ginsparg-Wilson

(GW) relation [2] and therefore have an exact chiral symmetry which can be identiﬁed with the ax-

ial U(1) symmetry of continuum massless Dirac fermions [3]. The latter is broken by quantum

effects – speciﬁcally by the fermion measure á la Fujikawa [4] – so there is an axial anomaly. It is

proportional to an index density for the index of the overlap Dirac operator, just like in the contin-

uum. (The continuum anomaly is proportional to the topological charge density, whose integral –

the topological charge of the background gauge ﬁeld – is equal to the index by the Index Theorem.)

An important test of the overlap lattice fermion formulation is then to show that the continuum

anomaly and index are reproduced in the continuum limit in smooth gauge ﬁeld backgrounds. This

has been veriﬁed in a number of papers at various levels of rigor [2, 5, 6, 7], and also in a more

general setting where the kernel operator in the overlap formulation is a more general hypercubic

lattice Dirac operator [8].

The advantageous theoretical properties of overlap fermions are offset by the high cost of im-

plementing them in numerical simulations of lattice QCD. Recently, a staggered version of the

overlap fermions, describing 2 fermion species (ﬂavors), was introduced in [9] as a further devel-

opment of the spectral ﬂow approach to the staggered fermion index in [10]. The fermion ﬁeld in

this case is a one-component ﬁeld (no spinor indices) just like for staggered fermions. This offers

the prospect of theoretical advantages of overlap fermions at a cheaper cost.

1

To establish a secure theoretical foundation for staggered overlap fermions, one of the things

that needs to be done is to verify that it reproduces the continuum anomaly and index in the con-

tinuum limit in smooth gauge ﬁeld backgrounds, just like for usual overlap fermions. The purpose

of this paper is to sketch the veriﬁcation of this. Full details will be given in a later article.

2

There

are new complications for evaluating the continuum limit of the anomaly and index in this case

compared to the usual overlap case, due to the fact that the spin and ﬂavor components are dis-

tributed around the lattice hypercubes in the staggered formalism. It turns out that, although the

index correctly reproduces the continuum index, the axial anomaly only reproduces the continuum

anomaly after averaging over the sites of a lattice hypercube. This is not surprising, since the sites

around a lattice hypercube can be regarded as corresponding to the same spacetime point in the

staggered formalism.

We focus on the original 2-ﬂavor version of staggered overlap fermions introduced in [9].

A 1-ﬂavor version of staggered Wilson fermions was later introduced in [14] and can be used as

kernel to make a 1-ﬂavor version of staggered overlap fermions. However, this formulation has

the drawback of breaking lattice rotation symmetry; a new gluonic counterterm then arises which

needs to be included in the bare action and ﬁne-tuned to reproduce continuum QCD, thus reducing

the attractiveness of this formulation for practical use. Nevertheless, it correctly reproduces the

1

An exploratory investigation of the cost of staggered overlap fermions was made in [11]. More recently, the cost of

staggered Wilson fermions was investigated in [12]; the results there give positive indications for the cost effectiveness

of staggered overlap fermions on larger lattices.

2

Numerical checks that the staggered overlap Dirac operator has the correct index in smooth backgrounds have

already been done in [13] (for lattice transcripts of instanton gauge ﬁelds) and in [11].

2

Axial anomaly and index for the staggered overlap Dirac operator Reetabrata Har

continuum axial anomaly and index; this can be shown by a straightforward modiﬁcation of our

arguments here for the 2-ﬂavor case; the details will be given in the later article.

2. The staggered overlap Dirac operator, its index, and the axial anomaly

The staggered overlap Dirac operator D

so

is given by

D

so

=

1

a

1+ A/

√

A

†

A

, A = D

sW

−

m

a

(2.1)

where D

sW

is the staggered Wilson Dirac operator, obtained by adding a “Wilson term” to the

staggered fermion action as discussed below (see (3.2)). The staggered Wilson term reduces the

number of fermion species (ﬂavors) described by the staggered fermion from 4 to 2 (the other 2

species get masses ∼1/a and become doublers) [9]. Consequently, for suitable choice of m in (2.1),

the staggered overlap fermion describes 2 physical fermion ﬂavors. Speciﬁcally, the requirement is

0 < m < 2 just like for usual overlap fermions.

The role of unﬂavored

γ

5

in this setting is played not by the unﬂavored

γ

5

of the staggered

fermion but by the ﬂavored

γ

5

which gives the exact ﬂavored chiral symmetry of the staggered

fermion action, which we denote by Γ

55

. This notation reﬂects the fact that it corresponds to

γ

5

⊗

ξ

5

in the spin-ﬂavor interpretation of staggered fermions by Golterman and Smit [15], where

{

ξ

µ

} is a representation of the Dirac algebra in ﬂavor space.

3

The use of Γ

55

as the unﬂavored

γ

5

in the staggered versions of Wilson, domain wall and overlap fermions is justiﬁed by the fact that

it acts in a unﬂavored way on the physical fermion species, since

ξ

5

= 1 on these [9].

With Γ

55

playing the role of unﬂavored

γ

5

, the requirements

γ

2

5

= 1 and

γ

5

hermiticity of the

staggered Wilson Dirac operator D

sW

are satisﬁed, and hence the staggered overlap Dirac operator

satisﬁes the the Ginsparg-Wilson (GW) relation [2], which in this case is

Γ

55

D

so

+ D

so

Γ

55

= aD

so

Γ

55

D

so

. (2.2)

Moreover, (2.1) can be rewritten as

D

so

=

1

a

1+ Γ

55

H/

√

H

2

(2.3)

where

H = H

sW

(m) = Γ

55

(D

sW

−m/a) (2.4)

is a hermitian operator.

The index of D

so

is then determined by the spectral ﬂow of H

sW

(m) just like in the usual

overlap case, and can be computed from the index formula in the 2nd paper of [2]:

indexD

so

= −

1

2

Tr

H/

√

H

2

. (2.5)

Here Tr is the operator trace for operators on the space of lattice staggered fermion ﬁelds, i.e.

functions living on the lattice sites and taking values in the fundamental representation of the color

gauge group SU(3) (color indices but no spinor indices).

3

Γ

55

is denoted by

ε

in [15].

3

Axial anomaly and index for the staggered overlap Dirac operator Reetabrata Har

The lattice spacetime is the usual hypercubic discretization of a 4-dimensional box of ﬁxed

length L, with N sites along each axis so that the lattice spacing is a = L/N. The link variable

associated with a lattice link [x, x + a

µ

] is denoted U

µ

(x) or U(x,x + a

µ

). (The index

µ

is also

used to denote the unit vector along the positive

µ

-axis.) The link variables are taken to be the

lattice transcripts of a smooth continuum gauge ﬁeld A

µ

(x) taking values in the Lie algebra of

SU(3). Boundary conditions are imposed as in [7] by requiring ﬁelds at opposite ends of the box to

be related by gauge transformations in such a way that A

µ

(x) can be topologically nontrivial with

topological charge Q ∈Z (see [7] for the details). The box contains ﬁnitely many lattice sites, so the

vector space of lattice fermion ﬁelds is ﬁnite-dimensional and hence the index (2.5) is well-deﬁned

for all choices of m for which H

sW

(m) does not have zero-modes. In the following we restrict for

simplicity to the physical choices where 0 < m < 2.

From (2.5) the index can be expressed as a sum over lattice sites (“lattice spacetime integral”)

of a density q(x):

indexD

so

=

∑

x

a

4

q(x) , q(x) = −

1

2

tr

c

H

√

H

2

(x,x) (2.6)

where tr

c

denotes the trace over color indices. Here we are using the concept of operator density

O(x,y), deﬁned by O

χ

(x) =

∑

y

a

4

O(x,y)

χ

(y), which can be used to express the operator trace as

TrO =

∑

x

a

4

tr

c

O(x,x).

The index density q(x) is seen to be closely related to the axial anomaly in the same was as

for usual overlap fermions [3]: The staggered overlap action

¯

χ

D

so

χ

is invariant under the axial

U(1) transformations

δχ

= Γ

55

(1−aD

so

)

χ

,

δ

¯

χ

=

¯

χ

Γ

55

. The corresponding axial current fails to

be conserved: its divergence fails to vanish, and is found in the massless limit to be A (x) = 2iq(x),

which is by deﬁnition the anomaly.

In the usual overlap case, the main step in showing that both the index and axial anomaly

have the correct continuum limits is to show that the index density q(x) reproduces the continuum

topological charge density [6, 7]:

lim

a→0

q(x) = −

1

32

π

2

ε

µνσρ

tr

c

(F

µν

(x)F

σρ

(x)) (2.7)

Here x is ﬁxed point in the spacetime box, and the a → 0 limit is taken by repeated subdivisions of

the lattice such that x continues to be a lattice site of all the subdivided lattices. However, in the

present case, it turns out that (2.7) only holds after q(x) is averaged over the sites of a lattice hy-

percube containing x. Note that the sites of the hypercube all converge to x in the continuum limit,

so this situation is not unreasonable, and is moreover not surprising in the staggered formalism as

mentioned in the Introduction.

Our goal in the remainder of this paper is to sketch the derivation of the modiﬁed version of

(2.7), i.e. with q(x) averaged over the sites of a lattice hypercube containing x, and with an extra

factor of 2 due to the two physical fermion ﬂavors described by the staggered overlap fermion.

A further technical result is required to conclude that the index expression (2.6) converges to

the continuum topological charge 2Q (which equals the continuum Dirac index with 2 ﬂavors by

the continuum index theorem): It needs to be shown that the convergence in (2.7) is uniform in x.

This can be shown in the same way as in the usual overlap case in [7]; we omit the details here.

4

Axial anomaly and index for the staggered overlap Dirac operator Reetabrata Har

3. Continuum limit of the lattice index density

By the same arguments as in the usual overlap case [7] we have

lim

a→0

H

√

H

2

(x,x) = lim

a→0

Z

3

π

/2a

−

π

/2a

d

4

pe

−ipx

H

√

H

2

e

ipx

. (3.1)

Now decompose the lattice momentum p ∈[−

π

/2a,3

π

/2a]

4

as p =

π

a

A+q, where q ∈[−

π

/2a,

π

/2a]

4

and A = (A

1

,A

2

,A

3

,A

4

) with A

µ

∈ {0,1}. In the following, if A and B are two such vectors, we

also consider A + B to be a vector of this type with the components (A + B)

µ

∈ {0,1} mod 2.

Let V be the vectorspace spanned by the 16 plane waves {e

A

(x) ≡ e

i

π

a

Ax

}. In [15] two com-

muting representations of the Dirac algebra are deﬁned on V, namely {

ˆ

Γ

µ

} and {

ˆ

Ξ

ν

}, given by

(

ˆ

Γ

µ

)

AB

= (−1)

A

µ

δ

A,B+

η

µ

and (

ˆ

Ξ

ν

)

AB

= (−1)

A

µ

δ

A,B+

ζ

ν

where

η

µ

and

ζ

ν

are vectors whose com-

ponents are all zero except (

η

µ

)

σ

= 1 for

σ

<

µ

and (

ζ

ν

)

σ

= 1 for

σ

>

ν

.

4

The staggered Wilson Dirac operator [9] is D

sW

= D

st

+W

st

where D

st

is the usual (mass-

less) staggered Dirac operator and W

st

=

r

a

(1 −Ξ

5

) is the Wilson term. Here Ξ

5

(= Γ

55

Γ

5

) is

an extension of

ˆ

Ξ

5

from V to the space of staggered lattice fermion ﬁelds

χ

(x), described in [9].

For present purposes it sufﬁces to note that the action of the staggered Wilson Dirac operator on

a plane wave e

ipx

can be expressed as follows. Let T

+

µ

denote the parallel transporter given by

T

+

µ

χ

(x) = U(x,x + a

µ

)

χ

(x + a

µ

), and let T

−

µ

denote its inverse. The symmetrized covariant

derivative is ∇

µ

=

1

2

(T

+

µ

−T

−

µ

), and we deﬁne C

µ

=

1

2

(T

+

µ

+ T

−

µ

). We will also need the sym-

metrized product of the C

µ

s, namely C

5

≡

1

4!

∑

C

µ

C

ν

C

σ

C

ρ

where the sum is over all permutations

{

µ

,

ν

,

σ

,

ρ

} of {1,2,3,4}. Then, writing the plane wave as e

ipx

= e

B

(x)e

iqx

, the action of D

sW

on

it can be expressed as

D

sW

(e

B

(x)e

iqx

) = e

A

(x)

(

ˆ

Γ

µ

)

AB

∇

µ

+

r

a

δ

AB

−(

ˆ

Ξ

5

)

AB

C

5

e

iqx

= e

A

(x)

ˆ

Γ

µ

∇

µ

+

r

a

1−

ˆ

Ξ

5

C

5

AB

e

iqx

≡ e

A

(x)(

˜

D

sW

)

AB

e

iqx

(3.2)

Here and in the following there is an implicit sum over repeated indices. This includes the vector

A which is regarded as an index with 16 possible values.

Since H is built from D

sW

and Γ

55

, and the latter acts on plane waves by Γ

55

(e

B

(x)e

iqx

) =

e

A

(x)(

ˆ

Γ

5

ˆ

Ξ

5

)

AB

e

iqx

, it follows that the action of H/

√

H

2

on a plane wave has the same structure as

in (3.2):

H

√

H

2

(e

B

(x)e

iqx

) = e

A

(x)

˜

H

√

˜

H

2

AB

e

iqx

(3.3)

where

˜

H is obtained from H by replacing D

sW

→

˜

D

sW

and Γ

55

→

ˆ

Γ

5

ˆ

Ξ

5

.

Using the preceding to evaluate (3.1), we get

lim

a→0

H

√

H

2

(x,x) = lim

a→0

∑

B

Z

π

/2a

−

π

/2a

d

4

qe

−iqx

e

B

(x)

H

√

H

2

e

B

(x)e

iqx

= lim

a→0

∑

A,B

e

i

π

a

(A−B)x

Z

π

/2a

−

π

/2a

d

4

qe

−iqx

˜

H

√

˜

H

2

AB

e

iqx

(3.4)

4

The hats on

ˆ

Γ

µ

and

ˆ

Ξ

ν

are not present in the notation of [15]. We include them here, since we use the unhatted

versions to denote the extensions of these operators from V to the space of one-component lattice spinor ﬁelds

χ

(x).

5

Axial anomaly and index for the staggered overlap Dirac operator Reetabrata Har

We will show below that (i) the contribution to q(x) from the terms in (3.4) with A = B reproduces

the continuum topological charge density (2.7), and (ii) the contributions from the terms with A 6= B

vanish after averaging q(x

′

) over the sites x

′

in a lattice hypercube containing x. This will complete

the derivation of the averaged version of (2.7).

The sum over A,B with A = B in (3.4) gives

lim

a→0

Z

π

/2a

−

π

/2a

d

4

qe

−iqx

Tr

˜

H

√

˜

H

2

e

iqx

(3.5)

where Tr denotes the trace of a linear operator (matrix) on the vectorspace V spanned by the e

A

(x)’s.

As shown in [15], there is a (non-unique) isomorphism V ≃ C

4

⊗C

4

such that

ˆ

Γ

µ

≃

γ

µ

⊗1 and

ˆ

Ξ

ν

≃1⊗

ξ

ν

, where {

γ

µ

}and {

ξ

ν

}are Dirac matrices on spinor space and ﬂavor space, respectively.

After choosing a basis for ﬂavor C

4

such that

ξ

5

is diagonal, (3.2) gives

˜

D

sW

= (

γ

µ

⊗1)∇

µ

+ (1⊗1)

r

a

(1∓C

5

) (3.6)

with sign ∓ for the ﬂavors on which

ξ

5

= ±1. Thus, on each of the two 2-dimensional ﬂavor

subspaces on which

ξ

5

= ± 1,

˜

D

sW

is a hypercubic lattice Dirac operator of the form considered in

[8] (note that C

5

couples opposite corners of lattice hypercubes). Also, Γ

55

becomes ±(

γ

5

⊗1) in

this case.

The free ﬁeld momentum representation of C

5

is C

5

(aq) =

∏

µ

cos(aq

µ

) = 1+ O(a

2

). Using

this, we see from (3.6) that

˜

D

sW

describes one physical fermion species for each of the two ﬂavors

of the ﬂavor subspace with

ξ

5

= 1. Now note that in this case (3.5) is precisely tr(H/

√

H

2

)(x,x) for

H = (

γ

5

⊗1)(

˜

D

sW

−m). Hence it is given by the general result of [8] on the continuum limit of the

anomaly and index for lattice hypercube fermions. In (3.5) the part of the trace Tr over the ﬂavor

subspace is trivial and just produces a factor 2, leaving the trace over spinor space. The general

result of [8] then implies that this contribution of (3.5) to q(x) gives the continuum topological

charge density (2.7) as claimed, with an extra factor of 2 for the two physical fermion ﬂavors.

On the other hand, the contribution of (3.5) coming from the other 2-dimensional ﬂavor sub-

space on which

ξ

5

= −1 vanishes. This can also be inferred from the general result of [8], since in

this case the sign ∓ is + in (3.6) and hence there are no physical fermion species.

It remains to show that the contributions to (3.4) from the terms in the sum over A,B with

A 6= B vanish when averaged over x ∈ {a lattice hypercube}. Writing x = an, n ∈ Z

4

, we have

e

i

π

a

(A−B)x

= (−1)

(A−B)n

. It is easy to see that summing this over the sites x of a lattice hypercube

gives zero if at least one of the components of A−B is nonzero (mod 2). Consequently, the problem

of showing that the hypercube-averaged terms with A 6= B in (3.4) vanish is reduced to showing a

property of the x-dependence of the integral there,

Z

π

/2a

−

π

/2a

d

4

qe

−iqx

˜

H

√

˜

H

2

AB

e

iqx

, (3.7)

namely, that this integral changes by O(a) as x is varied among the sites of a lattice hypercube.

The argument for this is as follows. Formally, (3.7) diverges ∼ 1/a

4

for a → 0. However, the

integrand can be expanded in powers of the continuum gauge ﬁeld just like in the usual overlap case

6

Axial anomaly and index for the staggered overlap Dirac operator Reetabrata Har

[6, 7]

5

and it can be shown that only the terms of mass-dimension 4 or higher in this expansion are

nonvanishing.

6

For each mass-dimension of the expansion terms there is an accompanying power

of a, so the nonvanishing terms contain at least a factor a

4

which balances the divergence ∼ 1/a

4

in (3.7). The fact that (3.7) changes by O(a) as x is varied among neighboring lattice sites then

follows from the fact that the smooth continuum gauge ﬁeld has this property.

Acknowledgments. D.A. is supported by AcRF grant RG61/10 and a start-up grant from

NTU.

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F. Niedermayer, Phys. Lett. B 427 (1998) 125 [hep-lat/9801021]

[3] M. Lúscher, Phys. Lett. B 428 (1998) 342 [hep-lat/9802011]

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[hep-th/9812019]

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Yukawa Institute Workshop “New Types of Fermions on the Lattice” Kyoto, 2012.

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5

This relies on a certain bound on the spectrum of the hermitian operator H when the plaquette variable of the

lattice gauge ﬁeld satisﬁes an approximate smoothness condition. It was established in the usual overlap case in [16].

An analogous bound holds in the present staggered overlap case; the proof will be given elsewhere.

6

This relies on two things: First, exponential locality of the integrand of (3.7) in the gauge ﬁeld (shown in the

same way as in the usual overlap case [16]), which implies that the terms in the expansion become local functionals

of the gauge ﬁeld in the a → 0 limit. Second, the gauge invariance and lattice rotation invariance of staggered overlap

fermions imply that the local functionals of the continuum gauge ﬁeld that arise in the expansion of (3.7) must have

mass-dimension ≥ 4. (There are no such terms with mass-dimension ≤ 3, while for mass-dimension 4 there are the

Yang-Mills action functional and topological charge density.)

7