The topological characteristics of lattice Dirac operators

We formulate the topological characteristics of lattice Dirac operators in the context of the index and the chiral anomaly, and illustrate this concept by explicit examples. PACS numbers: 11.15.Ha, 11.30.Rd, 11.30.Fs In continuum, the Dirac operator of massless fermions in a smooth background gauge field with nonzero topological charge Q has zero eigenvalues and the corresponding eigenfunctions are chiral. The Atiyah-Singer index theorem [1] asserts that the difference of the number of left-handed and right-handed zero modes is equal to the topological charge of the gauge configuration, n− − n+ = Q . (1) However, on the lattice, (1) may not be always well defined. Firstly, a lattice Dirac operator D may not have exact zero modes with definite chirality even if D is free of species doubling and agrees with γμ(∂μ + iAμ) in the classical continuum limit. Secondly, the topological charge may not be a well defined integer for any gauge link configurations. Nevertheless, for any smooth gauge configuration in continuum, we can construct the corresponding link configuration on a lattice such that the topological charge agrees with that in continuum. Then we can quest whether (1) holds on the lattice for such prescribed gauge backgrounds. With this provision, the assertion of (1) on the lattice only depends on the topological characteristics, c[D], of the lattice Dirac operator D. The formal definition of c[D] is n− − n+ = c[D] Q (2) where we have assumed that if D possesses exact zero modes, then they have definite chiralities. For topologically nontrivial sectors, it follows from (2) that in general c[D] is a rational number. For the vacuum sector ( Q = 0 ), c[D] can not be defined by (2), however, we can determine it unambiguously in the context of the chiral anomaly, as we will show below. In general, c[D] is a functional of D[U ] and it depends on the background gauge field nonperturbatively. In other words, even if D behaves correctly in the classical continuum limit, c[D] is not necessarily equal to one. Moreover, even if c[D] is equal to one for the vacuum sector, it may be zero for topologically nontrivial gauge fields. In a prescribed gauge background, we can classify any lattice Dirac operator according to its topological characteristics as follows : If c[D] = 1 ( c[D] = 0 ), D is called topologically proper ( topologically trivial ), otherwise D is called topologically improper. It is evident that we are only interested in those lattice Dirac operators which are topologically proper ( i.e., reproduce the Atiyah-Singer index theorem on the lattice ), at least for prescribed smooth gauge backgrounds. We assume that the lattice is finite with periodic boundary conditions, embedding in a d-dimensional torus. Examples of prescribed background gauge field are given in the Appendix of [2] 3 For any normalD ( i.e., DD = DD ) satisfying the hermiticity conditionD = γ5Dγ5, its zero modes have definite chiralities.