# Localization properties of lattice
fermions with

plaquette and improved gauge actions

###### Abstract

We determine the location of the mobility edge in the spectrum of the hermitian Wilson operator in pure-gauge ensembles with plaquette, Iwasaki, and DBW2 gauge actions. The results allow mapping a portion of the (quenched) Aoki phase diagram. We use Green function techniques to study the localized and extended modes. Where we characterize the localized modes in terms of an average support length and an average localization length, the latter determined from the asymptotic decay rate of the mode density. We argue that, since the overlap operator is commonly constructed from the Wilson operator, its range is set by the value of for the Wilson operator. It follows from our numerical results that overlap simulations carried out with a cutoff of 1 GeV, even with improved gauge actions, could be afflicted by unphysical degrees of freedom as light as 250 MeV.

###### pacs:

11.15.Ha, 12.38.Gc, 72.15.Rn## I Introduction

Domain-wall and overlap fermions reconcile chiral symmetry with the lattice, allowing for exact chiral symmetry at finite lattice spacing in the euclidean path-integral formulation dbk ; dwf ; oovlp ; ovlp ; GWL . While chiral symmetry can be achieved for a range of non-zero bare coupling , problems arise if the bare coupling is too large. For domain-wall fermions, chiral symmetry cannot be maintained in the strong-coupling limit BBS . For overlap fermions, the built-in (modified) chiral symmetry is exact, but at strong coupling one loses either locality HJL ; lclz or control over the number of species ovlp ; hopp .

In any numerical simulation
it is important to stay away from the dangerous regions of the phase diagram.
The lattice Dirac operators of domain-wall fermions (DWF)
and overlap fermions are both based on a Wilson operator with a negative,
super-critical bare mass .^{1}^{1}1
The super-critical region is .
Outside of this region the Wilson operator cannot
have zero eigenvalues.
Locality and chirality in these formulations are controlled
by the spectral properties of this underlying Wilson operator.

The outstanding features of the super-critical Wilson operator are best illustrated in a theory with two dynamical flavors of Wilson fermions. Here the absence of a spectral gap in part of the phase diagram implies the existence of propagating, light degrees of freedom. Moreover, a non-zero spectral density for vanishing eigenvalue signals spontaneous symmetry breaking, as follows from the Banks–Casher relation BC . For Wilson fermions there is no chiral symmetry to be broken, so the spontaneously broken symmetry is vectorial. As discovered by Aoki aoki , the pions become massless if the bare Wilson-quark mass is lowered from positive values towards a critical value . For the curvature of the effective potential for pions becomes negative at the origin; a pionic condensate forms which breaks spontaneously isospin and parity. This is the Aoki phase. Inside the Aoki phase the condensing pion is massive, while the other two pions are Goldstone bosons of the spontaneously broken isospin generators. See Fig. 1 for a schematic representation of the phase diagram.

For DWF or overlap fermions one aims for a big gap, of order , in the spectrum of the Wilson operator. Simply speaking, a bigger gap in the spectrum of the Wilson operator improves both the chiral symmetry of DWF (at fixed, finite extent of the fifth dimension) and the locality of overlap fermions. (As we will shortly see, in reality any such gap can be mostly, but not completely, devoid of eigenvalues.) Assuming that the phase diagram of the underlying Wilson operator remains qualitatively the same as in Fig. 1, this requires being inside one of the C phases. In practice, the rightmost C phase is used which, for , coincides with the interval . In this interval, one lattice domain-wall (or overlap) field gives rise to one quark in the continuum limit.

When one studies the spectral properties of the Wilson operator as a kernel for DWF or overlap fermions, one is in fact considering a quenched Wilson-fermion theory, because the Boltzmann weight is derived from a different fermion operator. Usually, quenching means leaving out the fermion determinant altogether, but we will also consider the more general sense that the fermion determinant is that of DWF or the overlap operator. In this paper, we will study the spectrum of the Wilson operator for a variety of pure-gauge theories, but we will take our results as indicative of what might arise in a theory with dynamical fermions.

The Goldstone theorem connects spontaneous symmetry breaking with the appearance of massless poles in correlation functions. Quenching the Wilson fermions, however, opens the door to a new dynamical possibility. Considering again a two-flavor theory, let us suppose that all the correlation functions of the two (quenched) Wilson flavors decay exponentially. (As we will see, this is indeed the case well inside the C phases.) In the quenched theory, this does not preclude a non-zero pion condensate: The isospin symmetry can be broken spontaneously without creating a Goldstone boson. It was shown in ms ; lclz how this can be reconciled with the usual Ward-identity argument for the existence of a Goldstone pole.

There is now solid numerical and semi-analytical evidence SCRI ; BNN for the existence of zero modes of the Wilson operator throughout practically the entire super-critical region. The Banks–Casher relation again leads to a non-zero pion condensate. In particular, the pion condensate is non-zero for parameter values in the C phase that yield good-quality DWF and overlap-fermion simulations. Despite the condensate, all the Wilson-fermion correlation functions are short ranged.

Ref. lclz, provides a theoretical explanation of this situation.
The low-lying eigenvectors of the hermitian Wilson operator may
be either extended or (exponentially) localized.
In the first case, the condensate must be accompanied by Goldstone bosons.
In the second case, there is an alternative mechanism for
saturating the relevant Ward identity, and all correlation functions
can be (and, in fact, are) short-ranged.
This gives the following physical picture
for the quenched Wilson-fermion phase diagram.^{2}^{2}2See also Ref. lt03, , in particular Fig. 2 therein.
In the entire super-critical region there is no gap
in the spectrum of the Wilson operator, and the pion condensate is non-zero.
For eigenvalues above a certain mobility edge ,
the eigenvectors
of the Wilson operator are extended.
If , eigenvalues
correspond to localized eigenmodes. In this case the pion condensate is non-zero, but
there are still no long-range correlations.
When , on the other hand, the condensate arises from extended eigenmodes,
and there are Goldstone pions.

The quenched Aoki phase is identified with the region where Goldstone pions exist; that is, it is defined by . With this definition, the quenched phase diagram could be qualitatively similar to that depicted in Fig. 1. Early numerical evidence supporting this quenched phase structure may be found in Ref. aokiq, . The weak-coupling region may also be studied via an effective lagrangian GSS .

As far as DWF and overlap fermions are concerned, the requirement of a gap in the Wilson spectrum should be replaced by the requirement that lclz . In other words, one must work outside of the (quenched) Aoki phase, in one of the C phases. It is therefore important to map out the Aoki phase on any ensemble used for DWF or overlap-fermion numerical simulations. Furthermore, for practical reasons, one should not be too close to the Aoki phase. How close is “too close” depends on the underlying Boltzmann weight and on the construction of the fermion operator. We will discuss this very practical point at some length in our conclusions.

In this paper we study the spectral properties of the Wilson operator via calculation of its resolvent and correlation functions derived from it. The theoretical framework developed in Ref. lclz, is directly applicable, and guides us in the numerical implementation. We measure the spectral density as well as properties that characterize the shape and size of the localized eigenmodes. The resolvent gives us these quantities much more economically than would the direct study of the eigenvalues and eigenvectors of . The correlation functions address the Ward identities and the Banks–Casher relation directly.

The resolvent allows simultaneous treatment of localized and extended modes. In any volume, the eigenvalues corresponding to localized modes are random. When the resolvent is averaged over the gauge ensemble, the single-configuration spectral density, , is smeared. Thus the ensemble-averaged spectral density is a continuous function of the eigenvalue for localized as well as extended modes. The essential physics of the localized modes lies not in their discreteness but in the compactness of their wave functions.

Our measurements are carried out on pure-gauge ensembles (which is the usual meaning of quenching). We compare the spectral properties for three different pure-gauge actions: the standard plaquette action and the Iwasaki iwa ; iwaval and DBW2 dbw2 ; dbw2val actions, two gauge actions motivated by renormalization-group considerations. These gauge actions have been used in quenched iwCPPACS ; dbw2RBC and dynamical ddwf DWF simulations, and in quenched overlap simulations 1gev ; gall .

This paper is organized as follows. In Sec. 2 we give basic definitions and derive the relevant Ward identities. In Sec. 3 we review the Banks–Casher relation as well as the localization alternative to Goldstone’s theorem. A twisted-mass term aoki ; tm provides the “magnetic field” that determines the direction of the pion condensate. Careful study of the vanishing twisted-mass limit reveals that, if the low-lying eigenmodes are localized, the two-point function of the would-be Goldstone pions diverges linearly with the inverse twisted mass. This enables the relevant Ward identity to be saturated without a Goldstone pole.

We then turn to our numerical investigations, starting with the standard plaquette action for the gauge field. In Sec. 4 we present results for the simplest quantity, the spectral density. In Sec. 5 we define the localization length and use it to determine the mobility edge for several points in the phase diagram. Extrapolations of to zero allow us to map out a part of the boundary of the Aoki phase. We then proceed to detailed study of the localized modes. In Sec. 6 we extend the investigation to the Iwasaki and DBW2 gauge actions. We conclude in Sec. 7 with a discussion of the implications of our results for domain-wall and overlap fermions. The results of Sec. 5 yield several quantities that help locate regions of the phase diagram to be avoided in simulations.

A concise account of this work, not including the improved gauge actions, has already been given mob .

## Ii Definitions

### ii.1 Fermion action

The Wilson–Dirac operator is defined as

(1) |

where

(2) |

comes from the naive Dirac operator, and

(3) |

is the Wilson operator that breaks chiral symmetry while preventing species doubling. Here , where are the three Pauli matrices; we are using a chiral basis for the Dirac matrices, where is diagonal. is the SU() matrix representing the gauge field. We study the spectrum of the hermitian Wilson–Dirac operator,

(4) |

The corresponding eigenvalue equation (in a given gauge field) is

(5) |

and we normalize the eigenvectors according to .

Previous studies of the spectrum of SCRI ; JLSS ; AT were based on the calculation of individual eigenfunctions and eigenvalues. We find it more economical to calculate the Green function

(6) |

where , in order to extract information about the spectrum. is well defined in finite volume provided . It has the spectral representation

(7) |

### ii.2 Two flavors and twisted mass

As mentioned in the Introduction, the spectral properties of have profound effects on the realization of continuous symmetries when there is more than one flavor. Thus we will add an isospin index to the fermion field and consider the two-flavor theory defined by

(8) | |||||

where . Spontaneous breaking of the flavor symmetry (and of parity) will be connected with the condensation of the “pion” field,

(9) |

where . The parameter has been introduced into Eq. (8) in order to shift the focus from zero to nonzero eigenvalues of . In order to control the isospin orientation of the condensate, we add to the action a “magnetic field” in the guise of a twisted-mass term, giving finally

(10) | |||||

will be used as a regulator to avoid the singularities of along the real axis.

### ii.3 SU(2) flavor symmetry and Ward identities

For , the fermion action (10) has a (vector) SU(2) flavor symmetry. For , the Ward identity of the broken symmetry is obtained by performing a local flavor transformation,

(11) |

and similarly for , where

(12) |

We find for any operator that

(13) |

Here the backward lattice derivative is defined by , and the vector current corresponding to Eq. (12) is

(14) |

While the notation indicates an integration over both fermion and gauge fields, the Ward identity (13) in its various guises is in fact valid for each gauge configuration separately.

## Iii Goldstone’s Theorem and localization

The Ward identity (20) is valid for arbitrary and . In a quenched theory, however, despite the Goldstone Theorem, does not necessarily imply the existence of a massless pole in in the limit . Let us recall lclz ; ms how this comes about.

### iii.1 Condensate and Banks–Casher relation

The volume-averaged pion condensate

(21) |

in the two-flavor theory (10) can be expressed in terms of the Green function . We will denote the expectation value in a given gauge field by . Then

(22) | |||||

where we have used the spectral representation (7). Averaging this over the gauge field gives the translation-invariant result

(23) |

where is the eigenvalue density defined by

(24) |

In the limit , we obtain

(25) |

This is a generalized Banks–Casher relation; the original relation BC is Eq. (25) at .

### iii.2 Localization as an alternative to Goldstone’s theorem

Naively taking the limit in Eq. (20) gives

(26) | |||||

where the second line is the approximate form for . As we shall see shortly, Eq. (26) is sometimes false, but it contains the Goldstone Theorem: must have a massless pole for any such that . By the generalized Banks–Casher relation (25), this happens whenever . Apart from , one expects long-range power-law decay also for other correlation functions, including in particular .

In the physics of disordered systems DJT
it is well known that the eigenmodes of a hamiltonian in a random background
divide into two classes: extended and localized.
In fact, the spectrum splits into bands,
each band containing only eigenmodes of one type.^{3}^{3}3There seems to be no rigorous proof of this
fact except in one dimension DJT .
A point in the spectrum separating an extended band from
a localized band is a mobility edge.
will exhibit a power-law decay when the lies in
an extended band,
while for a localized band it will decay exponentially.
We expect that the same basic separation applies to as well.

If comes from localized eigenmodes and has no pole at zero momentum, what has become of the Goldstone Theorem? In other words, how is the Ward identity (20) satisfied? The way out of this conundrum is the following. In the limit , diverges as for a range of values of that includes the point . The limiting value of is finite. For , we arrive at an alternative to Eq. (26),

(27) |

### iii.3 Divergence of the pion two-point function

Let us consider further the divergence in . The (finite-volume) spectral representation of the charged-pion two-point function is

(28) |

where terms with the subscript are associated with the propagator for the corresponding quark flavor. As explained in Sec. 3 of Ref. lclz, , a divergence may arise only from the terms with , so that

(29) |

In analogy with Eq. (24), we define the eigenmode-density correlation function,

(30) |

and its Fourier transform,

(31) |

where

(32) |

As , these may be interpreted as the contribution to , or to its Fourier transform, of eigenmodes with eigenvalue . Repeating the analysis leading from Eq. (22) to Eq. (25) we find, for ,

(33) |

Observe that is the ensemble average (30) of a quantity that is strictly positive. Also, since the eigenmodes are normalized, we have that

(34) |

and so means that must be nonzero for at least one pair of values . Consequently, in any finite volume, implies the existence of a divergence in the coordinate-space two-point function. [This must be true for at least one value of , but is expected to hold for practically every .] This result is valid for the generalized quenched theory defined by Eq. (10) for any Boltzmann weight; the only assumption we used is that the Boltzmann weight does not depend on and .

[An unquenched theory with fermion action (10) would include in the Boltzmann weight. This determinant will suppress eigenvalues , and the spectral density measured by Eq. (25) will be zero in any finite volume in the limit .]

We next consider the infinite-volume limit. The asymptotic behavior of an exponentially localized eigenmode is

(35) |

where is the localization length and is the center of the localized eigenmode. Eq. (35) is valid only at distances that are large compared to the size of the region containing most of the eigenmode’s density. In principle, nothing forbids the occurrence of eigenmodes with a very short localization length .

An extended eigenmode is one that does not satisfy Eq. (35) for any finite . Evidently, truly extended eigenmodes exist only in infinite volume. In finite volume, the clear-cut identification of an eigenmode as localized demands that is exponentially small on most of the lattice. In later sections we will give a more quantitative criterion.

In infinite volume, we expect the Fourier transform of the eigenmode density (35) to have the following small- behavior:

(36) |

The region will reflect only the exponentially decaying envelope (35) of the eigenmode density but not the short-distance fluctuations. The overall normalization is set by for a normalized eigenmode. Substituting the ansatz (36) into Eq. (31) and going to small we find, in analogy with Eq. (33),

(37) |

where is some average localization length for the eigenmodes with eigenvalue . [This extends Eq. (27) to nonzero .] From Eq. (20) we conclude that

(38) |

Thus there is no Goldstone pole when the eigenmodes with eigenvalue
are exponentially localized.^{4}^{4}4See also
Sec. 4 of Ref. lclz, .

When the eigenmodes at the given are extended, the transition from Eq. (28) to Eq. (29) is not justified if the limit is taken after the infinite-volume limit, because of interference effects between eigenmodes with infinitesimally close eigenvalues. In this case we expect Eq. (26) to be valid, indicating a Goldstone pole.

## Iv Wilson plaquette action: spectral density

We now turn to our numerical investigation. We begin with quenched ensembles generated with the Wilson plaquette action for the SU(3) gauge theory,

(39) |

We began calculating at , which is usually taken to correspond to
the lattice scale GeV, and at^{5}^{5}5For the remainder of the paper we rescale ,
giving the bare Wilson mass in lattice units.
,
between the Aoki “fingers” that point to the
() axis at and .
Then we moved downward in towards the Aoki phase, calculating
at , 5.7 (where GeV), 5.6, 5.5, and 5.4.
We will show that the Aoki phase is entered just below
(see Fig. 1).

Returning to , we moved “sideways” by changing to and . The latter turns out to be very close to or in the second Aoki finger. We put these choices of into the context of other work in Sec. VII.

For each value of we generated 120 uncorrelated gauge configurations (except where otherwise noted) on a lattice of sites, using the MILC pure-gauge overrelaxation code. For a given , results for all values of , , and were calculated on the same ensemble; thus correlations had to be taken into account in all fits and statistical analysis.

We shall illustrate our methods by discussing in detail the analysis for and . Results for other values of will be summarized in the tables.

### iv.1 Spectral density from Banks–Casher relation

From the Banks–Casher relation (25) and Eq. (22) we have

(40) |

Thus the volume- and ensemble-averaged Green function, extrapolated to , gives directly. Of course must be kept nonzero for actual calculation in order for to be bounded.

The spectral sum (22) shows how to do the extrapolation. For any gauge configuration we consider the sum

(41) |

The summand tends to a -function as , but before the limit is taken it has a finite width equal to . The given configuration will make a contribution of if has an eigenmode whose eigenvalue satisfies ; these contributions, summed over configurations , will add up to a finite limit as . On the other hand, all the eigenmodes that are far from , with , will make a contribution of . This indicates a linear extrapolation,

(42) |

where will depend on . Then is an estimate for .

We calculated using a single random source per gauge configuration. Averages were obtained for up to seven values of : 0.01, 0.02, , 0.07. The upper graph in Fig. 2 shows the linear extrapolation for two values of .

For the fit works well. For , on the other hand, the extrapolated intercept is very small and the precision attained is inadequate.

The problem is that, when is too small, there are few eigenmodes (for the volume we use) within the broadened -function in Eq. (41). Then the sum is dominated by the contributions of the more distant eigenmodes, with large fluctuations. Making even smaller will suppress these contributions, but it will be self-defeating because even fewer modes will lie within the broadened -function, increasing the fluctuations in their contribution as well.

### iv.2 Improved estimator

A solution lies in using an improved estimator for the spectral density, one that suppresses the contribution of distant eigenmodes and thus approaches the limit faster than linearly. We introduce the (dimensionless) differential operator defined by

(43) |

for any . It is designed to remove the linear term in . Applying it to the pion condensate, we have

(44) |

with the spectral representation

(45) |

The limit again yields since .

The operator indeed removes the leading, linear contribution of the eigenmodes with . We can see from Eq. (45) that the contribution of these eigenmodes is now proportional to . In view of this, we attempt a cubic extrapolation,

(46) |

The lower graph in Fig. 2 shows the new extrapolation, again at . Linear and quadratic terms are in fact unnecessary to attain an excellent fit to the data points; this is the final justification of the model (46). Most important, the precision in the extrapolation is improved by a factor of ten.

Using the improved estimator (44) requires a second inversion for each random source. The number of CG iterations needed for the second inversion can reach twice the number for the first inversion. The cost of the improved estimator is thus up to (roughly) three times the original cost. Were we to invest the additional computer time instead in increased statistics using the simple estimator (40), the anticipated reduction in the statistical error would not come anywhere close to the factor of ten achieved with the improved estimator.

### iv.3 Spectral density from the two-point function

Since the two-point function is the eigenmode-density correlator [see Eq. (29)], it is a rich source of information about spectral properties. The first application we discuss is an alternative method to calculate the spectral density .

We project to zero spatial momentum and calculate the time-correlation function,

(47) |

where is the three-volume. This calculation requires a random source on time slice 0, as well as a random sink for each time slice ; again we use one set of random sources per gauge configuration. If we sum over all , we can use translation invariance and Eq. (29) to show that

(48) |

This corresponds to setting in the finite-volume Ward identity (20). [Note the similarity to Eq. (34).] We verified Eq. (48) for two values of the bare coupling ( and 5.7) by calculating both the condensate [Eq. (40)] and the two-point function. In all subsequent measurements we did not calculate the condensate separately since it gives only the spectral density, while yields this and much more.

The extrapolation of to suffers from the same fluctuation problems as that of when the spectral density is small. The solution again lies in an improved estimator, but now for the eigenmode-density correlation function [see Eq. (30)]. We define the “improved” two-point function,

(49) |

where

(50) | |||||

and project it to zero spatial momentum,

(51) |

Instead of Eq. (29) one has

(52) |

whence

(53) |

Equation (34) then yields the spectral density [cf. Eq. (48)],

(54) |

Results obtained using were extrapolated to
linearly in ,
while those
obtained using were extrapolated^{6}^{6}6The form of Eq. (52) suggests an term as well,
but we found it to be unnecessary for a good fit.
linearly in .

We also tried to improve the estimator for the eigenmode density by calculating [cf. Eq. (43)]. This has the advantage that the application of to all terms in the Ward identity (20) generates a new identity. As it turns out, while some reduction in the statistical error was achieved this way, the improvement was not significant. The reason is that the application of to would be effective in suppressing the contribution of eigenmodes with only if the approximation Eq. (29) is valid. As it turns out, for the values we used, Eq. (29) is not a good approximation to Eq. (28) when the spectral density is small. The correlator does a better job in suppressing the contribution of eigenmodes with .

### iv.4 Numerical results

We present our results for the spectral density for the Wilson plaquette action in Tables 1–8 and in Figs. 3 and 4.

At each value of and , we performed our calculations for a range of values of starting from zero, increasing in steps of 0.1. In the ranges studied (the maximal was 0.4 to 0.6) the spectral density typically increases by one to three orders of magnitude. For the ensemble generated by the plaquette action, shows no remarkable behavior as passes , the mobility edge (to be determined below). In all cases, however, the mobility edge is encountered for . There is a steady rise in the spectral density as we decrease (Fig. 3), which bespeaks an increased disorder in the gauge field.

A final note on improvement: For the larger values of it is possible to achieve acceptable precision without using the improved estimator. Since the number of CG iterations required grows rapidly with the spectral density, we limited the use of improved estimators to those cases where results obtained without improvement were poor.

## V Wilson plaquette action: localization properties

A central goal of the work presented here is the determination of the mobility edge at various places in the phase diagram. Equally interesting is the shape and density of the localized eigenmodes below (where ).

The two-point function contains complete information about the eigenmodes of , and it is our primary tool both in determining and in studying the localized modes. When dealing with localized eigenmodes, we aim for parameter values where the single spectral sum (29) provides a good approximation to the exact expression (28). When is small enough, then reduces to the eigenmode-density correlator [Eq. (30)]. Our analysis is based on this feature. The approximation (29) ceases to hold when the eigenmodes become too dense or too extended, that is, when they interfere in the double sum. This occurs when is above or too close to the mobility edge. We will develop a criterion to establish the consistency of our analysis of the localized eigenmodes.

We begin by giving a precise definition of the localization length . Its divergence marks the mobility edge . By definition, marks the Aoki phase. We then turn to study other properties of the localized eigenmodes outside the Aoki phase, where .

### v.1 The localization length

We have loosely defined the localization length of an individual eigenmode in Eq. (35). We can use , nebulous as it is, to motivate the definition of an average localization length that can be obtained from from the large- behavior of . Once we reach this definition, a precise definition of will be superfluous.

We begin by introducing the restricted spectral density that contains the contributions of eigenmodes with localization length only. It is given by

(55) |

and its derivative, the differential spectral density, is given by

(56) |

We also introduce a probability distribution for the localization length of eigenmodes with given eigenvalue by writing . This distribution is normalized because, in finite volume,

(57) |

where is the (largest) linear size of the lattice. Below, this upper limit will be implicit.

[In infinite volume there can be truly extended eigenmodes. Since we expect the eigenmodes at a given to be either all extended or all localized, we have correspondingly either for all finite , or .]