FUNCTIONAL INTEGRATION AND QUANTUM MONTE CARLO IN ITINERANT MAGNETS: THE BERRY PHASE AND THE SIGN PROBLEM

Abstract

Itinerant magnets are widely studied by functional integration: a stochastic field {Δi(τ)} replaces the interaction in the Heisenberg and N-band Hubbard models, giving the partition function Z=ʃD3Δexp(–NβV[{Δi(τ)}]) as an integral over a time-dependent field. A difficulty in quantum Monte Carlo simulations is that some paths contribute negative weight to the integral. We consider the contribution exp(−NβV[{Δi(τ)}]) of an arbitrary path. The path has a phase, tending to a Berry phase for smooth paths in the low-temperature narrow-band limit (where spins follow fields adiabatically), and to zero in the high-temperature limit, recovering the static approximation. Time dependence can be integrated out in simple cases, yielding an effective static Hamiltonian Veff({Δi}). The weight exp[−NβVeff({Δi})] is real but not positive-definite.