# Ginsparg-Wilson relation with

###### Abstract

The Ginsparg-Wilson relation with is discussed. An explicit realization of is constructed. It is shown that this sequence of topologically-proper lattice Dirac operators tend to a nonlocal operator in the limit . This suggests that the locality of a lattice Dirac operator is irrelevant to its index.

## 1 Introduction

In general, the Ginsparg-Wilson relation [1] for a lattice Dirac operator can be written as

(1) |

where and are analytic functions. Then the fermionic action is invariant under the generalized chiral transformation

(2) | |||||

(3) |

where is a global parameter. In particular, for and , (1) becomes

(4) |

where is any operator. Since commutes with , without loss, we can set to commute with ,

(5) |

Then (4) becomes the usual form of the GW relation

(6) |

In the continuum, the massless Dirac operator is chirally symmetric and anti-hermitian, i.e.,

(7) | |||

(8) |

which imply that is -hermitian, i.e., .

On the lattice, a lattice Dirac operator cannot satisfy (7) or (8) without losing its other essential properties. However, we can still require to satisfy the -hermiticity,

(9) |

Multiplying on both sides of (6), and using (9), we obtain , which gives .

Thus we conclude that, in general, is a Hermitian operator commuting with . From (6), the GW fermion propagator ( in the trivial sector ) satisfies , which immediately suggests that should be as local as possible, i.e., . Otherwise, it may cause additive mass renormalization to , which of course is undesirable. In fact, it has been demonstrated that the most optimal is with some constant [2].

Nevertheless, theoretically, it is rather interesting to consider as a function of , and to see how behaves with respect to . It turns out that, in general, the independent variable in the functional form of must be , if is normal and -hermitian. In particular, for , the sequence of topologcially-proper (27) tend to a nonlocal Dirac operator in the limit . This suggests that the locality of a lattice Dirac is irrelevant to its index.

First, recall that for any which is normal
( ) and -hermitian,
its eigensystem ( )
has the following properties [3] :

(i) eigenvalues are either real or come in complex conjugate pairs.

(ii) the chirality ( )
of any complex eigenmode is zero.

(iii) each real eigenmode has a definite chirality.

(iv) sum of the chirality of all real eigenmodes is zero
( chirality sum rule ).

Assume that the eigenvalues of lying on a simple closed contour in the complex plane. Then (i) implies that the real eigenvalues of can only occur at two different points, say at zero and . Then the chirality sum rule reads

(10) |

which immediately gives

(11) |

where denotes the number of zero modes of chirality, and the number of nonzero real ( ) eigenmodes of chirality. Then we immediately see that any zero mode must be accompanied by a real ( ) eigenmode with opposite chirality, and the index of is

(12) |

It should be emphasized that the chiral properties (ii), (iii) and the chirality sum rule (11) hold for any normal satisfying the -hermiticity, as shown in Ref. [3]. However, in nontrivial gauge backgrounds, whether possesses any zero modes or not relies on the topological characteristics [11] of , which cannot be guaranteed by the conditions such as the locality, free of species doublings, correct continuum behavior, -hermiticity and the GW relation.

Therefore, we must require that is normal and -hermitian,

These two equations immediately give

(13) |

Thus we have found an example of which is Hermitian and commutes with . In general, can be any analytic function of , i.e.,

(14) |

In particular, for , it becomes

(15) |

Substituting (15) into the GW relation (6), we obtain

(16) |

which is equivalent to Fujikawa’s proposal [4]

(17) |

## 2 A Construction of

So far, the only viable way to construct a topologically-proper lattice Dirac operator is the Overlap [5]

(18) |

which satisfies the GW relation (6) with . Since (18) is normal and -hermitian, its eigenmodes satisfy the chiral properties (i)-(iv). There are many different ways to implement the Hermitian in (18). However, it is required to be able to capture the topology of the gauge background. That means, one-half of the difference of the numbers ( ) of positive ( ) and negative ( ) eigenvalues of is equal to the background topological charge ,

where tr denotes the trace over the Dirac and color space. Otherwise, the axial anomaly of cannot agree with the topological charge density in a nontrivial gauge background.

To generalize this construction for , we multiply both sides of the GW relation (6) by and redefine , then we have

(22) |

It can be shown that is -hermitian. Now (22) is in the same form of the GW relation with . Thus, one can construct in the same way as the Overlap

(23) |

provided that a proper realization of can be obtained. Using (5), we obtain

(24) |

which immediately yields

(25) |

where the -th real root of the Hermitian operator is assumed. Then (25) suggests that if the in (23) is expressed in terms of a Hermitian operator , i.e., , then is required to behave as in the continuum limit such that can become after taking the -th root in (25). Thus, must contain the term , where is the naive lattice fermion operator defined in (21). Then another term must be added in order to remove the species doublers in . So, we add the Wilson term to the -th power, i.e., . Finally, a negative mass term is inserted such that is able to detect the topological charge of the gauge background, i.e.,

(26) |

Putting all these terms together, we have

which, at , reduces to the in Eq. (20). Then (25) can be rewritten as

(27) |

which agrees with Fujikawa’s construction [4]. Note that (27) can also be written in the form , where is dependent on the order . The general properties of have been derived in [7]. In particular, the index of is independent of the order , and the eigenvalues of fall on a closed contour with their real parts bounded between zero and . In Fig. 1, the eigenvalues of are plotted for respectively, with the same background gauge field.

## 3 In the Limit

One of the salient features of the sequence of topologically-proper Dirac operators (27) is that the amount of chiral symmetry breaking ( i.e., r.h.s. of (6) ) decreases as the order increases. However, at finite lattice spacing, the chiral symmetry breaking of cannot be zero even in the limit , since satisfies the GW relation (6). In the limit , the only possibility for to break the chiral symmetry is that becomes a piecewise continuous function in the Brillouin zone, with discontinuities somewhere at , and for . Since such a is non-analytic at infinite number of , the corresponding must be nonlocal in the position space.

If is nonlocal in the free fermion limit ( i.e., ), then it must be nonlocal in any smooth gauge background. So, it suffices to examine the non-analyticity of in the free fermion limit.

In the free fermion limit, the GW Dirac operator (27) in momentum space can be written as

(28) |

where

At the zeroth order ( ), is analytic in the Brillouin zone. However, in the limit , tends to a nonanalytic function with discontinuities at each point on two concentric hypersurfaces inside the Brillouin zone [8]. In four dimensions, the inscribed hypersurface is specified by the equation

while the circumscribing one by

where and . This implies that is nonlocal in the limit .

On the other hand, is local at the zeroth order ( ), for gauge backgrounds which are sufficiently smooth at the scale of the lattice spacing [9, 10]. Since the index of (27) is independent of the order , this suggests that the locality of a lattice Dirac operator is irrelevant to its index. In other words, if a GW Dirac operator which is -hermitian, doubler-free, and has correct continuum behavior in the free fermion limit, but always has zero index [11], then the cause may not be due to its nonlocality.

To summarize, we have provided a nontrivial example of (27), which is topologically-proper but tends to a nonlocal operator in the limit . Unlike the nonlocal and chirally-symmetric Dirac operator

(29) |

which has poles ( i.e., the nonzero real eigenmodes ) for nontrivial background gauge fields ( as a consequence of the chirality sum rule [3] ), the present example (27) breaks the chiral symmetry according to the GW relation, thus its nonzero real ( ) eigenmodes are well-defined for any order .

## References

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