Simulating the all-order hopping expansion II: Wilson fermions

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$D=2$ Wilson fermions enjoy special positivity properties and the simulation is similarly efficient as in the Ising model. In D=3$D=3$ 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.