On the spectrum of the Wilson-Dirac lattice operator in topologically non-trivial background configurations

Abstract

We study characteristic features of the eigenvalues of the WilsonDirac operator in topologically non-trivial gauge field configurations by examining complete spectra of the fermion matrix. In particular we discuss the role of eigenvectors with real eigenvalues as the lattice equivalents of the continuum zero-modes. We demonstrate, that those properties of the spectrum which correspond to non-trivial topology are stable under adding fluctuations to the gauge fields. The behavior of the spectrum in a fully quantized theory is discussed using QED 2 as an example.

Spectrum of the Wilson Dirac operator at finite lattice spacings

“…As can be shown from spectral flow arguments (see, e.g. [21][22][23][24][25][26][27]), the eigenvalues corresponding to the chiral modes at zero lattice spacing (when the Dirac operator is antihermitian) correspond to exactly real eigenvalues at nonzero lattice spacing. Additional pairs of real modes appear for increasing lattice spacing.…”

General bounds on the Wilson-Dirac operator

“…For m ≤ 0 and m ≥ 2d (d=spacetime dimension) it is well-known that |H(m)| ≥ |m| and |H(m)| ≥ m−2d, respectively; see, e.g., [5,17,2]. (We review these bounds and generalise them to arbitrary values of the Wilson parameter r in the Appendix.)…”

“…A Bounds in the case of general Wilson parameter r > 0 Using (3.6)-(3.8) a simple calculation gives [5] H(m, r) 2…”

Fermions on tori in uniform Abelian fields

“…This has changed with the domain wall formulation of lattice fermions [9] and, especially, with the overlap formulation [10]. Among recent studies on the response of fermionic fields to topology in the overlap regularization we mention [10,11,12,13].…”