Index and overlap construction for staggered fermions

Abstract

Staggered fermions had long been perceived as disadvantaged compared to Wilson fermions

regarding the index theorem connection between (would-be) zero-modes and gauge field topology.

For Wilson fermions, the would-be zero-modes can be identified as eigenmodes with low-lying

real eigenvalues; these can be assigned chirality ±1 according to the sign of y¯ g5y, thereby determining

an integer-valued index which coincides with the topological charge of the background

lattice gauge field in accordance with the index theorem when the gauge field is not too rough

[1, 2, 3]. It coincides with the index obtained from the exact chiral zero-modes of the overlap Dirac

operator [4]. In contrast, for staggered fermions, no way to identify the would-be zero-modes was

known. They appeared to be mixed in with the other low-lying modes (all having purely imaginary

eigenvalues) [1, 5] and only separating out close to the continuum limit [6]. It seemed that, away

from the continuum limit, the best one could have was a field-theoretic definition of the staggered

fermion index [1]. The latter had the disadvantages of being non-integer, requiring a renormalization

depending on the whole ensemble of lattice gauge fields, and being significantly less capable

than the Wilson fermion index of maintaining the index theorem in rougher backgrounds