Simulating the All-Order Hopping Expansion II: Wilson Fermions

TL;DR

This paper extends a Monte Carlo algorithm to simulate Wilson fermions' hopping expansion on a lattice, enabling precise numerical tests and revealing dimensional differences in efficiency and sign problems.

Contribution

It introduces a novel Monte Carlo method for simulating Wilson fermions' hopping expansion, applicable in two and three dimensions, with detailed analysis of its efficiency and limitations.

Findings

Efficient simulation for D=2 Wilson fermions due to positivity properties.

Method works at large mass in D=3 but faces a sign problem.

Exact results allow rigorous testing of the numerical approach.

Abstract

We investigate the extension of the Prokof'ev-Svistunov worm algorithm to Wilson lattice fermions in an external scalar field. We effectively simulate by Monte Carlo the graphs contributing to the hopping expansion of the two-point function on a finite lattice to arbitrary order. Tests are conducted for a constant background field i. e. free fermions at some mass. For the method introduced here this is expected to be a representative case. Its advantage is that we know the exact answers and can thus make stringent tests on the numerics. The approach is formulated in both two and three space-time dimensions. In D=2 Wilson fermions enjoy special positivity properties and the simulation is similarly efficient as in the Ising model. In D=3 the method also works at sufficiently large mass, but there is a hard sign problem in the present formulation hindering us to take the continuum limit.