A Numerical Study of Spectral Flows of the Hermitian Wilson-Dirac Operator and the Index Theorem in Abelian Gauge Theories on Finite Lattices

163 Progress of Theoretical Physics, Vol. 107, No. 1, January 2002 A Numerical Study of Spectral Flows of the Hermitian Wilson-Dirac Operator and the Index Theorem in Abelian Gauge Theories on Finite Lattices Takanori Fujiwara∗) Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, Föhringer Ring 6, D-80805 München, Germany (Received January 16, 2001) We investigate numerically the spectrum of the hermitian Wilson-Dirac operator in abelian gauge theories on finite lattices. The spectral flows for a continuous family of abelian gauge fields connecting different topological sectors are shown to have a characteristic structure leading to the lattice index theorem. We find that the index of Neuberger’s Dirac operator coincides with the topological charge for a wide class of gauge field configurations. In two dimensions the eigenvalue spectra for some special but nontrivial configurations can be described by a set of characteristic polynomials and the index can be found exactly. It is generally believed that the nontrivial topological gauge field configurations are responsible for nonperturbative phenomena such as the large η-η  mass splitting in QCD and the fermion number violation in the Standard Model. Lattice gauge theories are the most promising approach to the study of such nonperturbative phenomena. However, the configuration space of lattice gauge fields is topologically trivial, since any configuration can be continuously deformed into the trivial one. This is not quite disappointing, because physically interesting gauge fields are smooth and we can impose a kind of smoothness conditions that restricts plaquette variables within a small neighborhood of unity. 1) The space of smooth link variables acquires a nontrivial topological structure and it is possible to define topological invariants geometrically in terms of lattice gauge fields. 1) - 4) The index theorem plays the role of a touchstone in lattice QCD. 5) - 8) In the case of Ginsparg-Wilson (GW) Dirac systems 9) - 11) it is possible to define exact chiral symmetry 12) and the index theorem on the lattice by relating the index of the GW Dirac operator to the chiral anomaly, as shown by Hasenfratz, Laliena and Niedermayer (HLN). 13), 12) Their index theorem is applicable not only for any gauge field configuration but also for any GW Dirac operator. Since the topological structure of the gauge fields is reflected by the construction of the GW Dirac operator implicitly, the HLN index theorem tells very little about the relationship between the index and the topological invariant of gauge fields for strictly finite lattices. Nevertheless, we expect that a lattice extension of the index theorem relating the index directly to the topological invariants of gauge fields can be established for sufficiently smooth configurations and physically acceptable GW Dirac operators. 14) In fact, it is possible to relate the axial anomaly (the index density) to the topological ∗) Permanent address: Department of Mathematical Sciences, Ibaraki University, Mito 310-8512, Japan. 164 T. Fujiwara charge density for abelian theories on infinite lattices quite generally by invoking general principles such as topological invariance, gauge invariance and locality. 15) On strictly finite lattices, however, we cannot use the known expressions of the chiral anomaly of the continuum theory to determine the overall normalization and, even worse, we cannot appeal to the locality. This makes the differential geometrical method developed in Ref. 15) inapplicable. Nevertheless, we expect that for sufficiently smooth gauge fields and for a sufficiently fine lattice, the index of the GW Dirac operator coincides with the topological charge.∗) To what extent the gauge fields should be smooth and to what extent the lattice should be fine to maintain the coincidence of the index and the topological charge are the main concern of the present paper. To obtain insight into these matters, detailed numerical work is necessary. In this paper, we focus our attention on the index of Neuberger’s Dirac operator for compact U (1) theories on finite periodic lattices in two and four dimensions and investigate the spectrum of the hermitian Wilson-Dirac operators numerically for a family of link variables connecting constant magnetic field configurations with distinct topological charges. In particular we are interested in the behavior of the spectrum as the gauge fields move from one topological sector to another. Such an analysis has already been reported by Narayanan and Neuberger 18) within the framework of the overlap formalism. They found characteristic behavior of the spectral flows leading to the index theorem. We extend their analysis concerning the spectral flows and the index theorem to include strong and nonsmooth gauge fields in the sense that they do not satisfy the smoothness conditions mentioned above. We investigate what happens on the spectral flows for gauge fields with large topological charges and how large the index may be for a given lattice size. We find that the index theorem is valid for a wider class of gauge fields than those satisfying the locality bounds. 14), 19) The coincidence of the index and the topological charge, however, no longer holds for configurations with large topological charges. We also mention some analytic results in two dimensions for the spectrum and the index. They are very useful to understand the numerical results. Let us begin with the definition of Neuberger’s Dirac operator on a d = 2N dimensional euclidean hypercubic regular lattice Ld with periodic boundary conditions. For simplicity we choose the lattice spacing as a = 1 and take the lattice size L to be a positive integer. We first introduce the hermitian Wilson-Dirac operator H by   Hψ(x) = γd+1 (d − m)ψ(x)    d   1 − γ 1 + γ µ µ  − Uµ (x)ψ(x + µ̂) + Uµ (x − µ̂)ψ(x − µ̂) , (1)  2 2 µ=1 ∗) For Neuberger’s overlap Dirac operator 11) the chiral anomaly on infinite lattices reduces to the well-known expressions of the continuum theory in the classical continuum limit. 16) (See also Ref. 17).) A Numerical Study of Spectral Flows 165 where m is the fermion mass, and the Wilson parameter r = 1 is assumed. The γ-matrices are taken to be hermitian and to satisfy {γµ , γν } = 2δµν . We use d γd+1 = (−i) 2 γ1 · · · γd . The link variables Uµ (x) are subject to the periodic boundary conditions Uµ (x + Lν̂) = Uµ (x). We also employ periodic boundary conditions for ψ(x).∗) Neuberger’s Dirac operator D is then given by H . D = 1 + γd+1 √ H2 (2) The value of m is chosen to satisfy 0 < m < 2 to avoid species doubling. We often use m = 1 in the following analysis and denote H(1) simply by H. Whenever the m dependence is relevant, we write it explicitly as H(m). For Neuberger’s Dirac operator (2), the index theorem of HLN can be stated as   1 1 H indexD = Trγd+1 1 − D  = − Tr √ . 2 2 H2 (3) This equation relates the index of D to the spectral asymmetry of H, since the trace on (...truncated)