Classical effective Hamiltonians, Wigner functions, and the sign problem
journal contributionposted on 2006-04-27, 13:14 authored by John Samson
In the functional-integral technique an auxiliary field, coupled to appropriate operators such as spins, linearizes the interaction term in a quantum many-body system. The partition function is then averaged over this time-dependent stochastic field. Quantum Monte Carlo methods evaluate this integral numerically, but suffer from the sign (or phase) problem: the integrand may not be positive definite (or not real). It is shown that, in certain cases that include the many-band Hubbard model and the Heisenberg model, the sign problem is inevitable on fundamental grounds. Here, Monte Carlo simulations generate a distribution of incompatible operators—a Wigner function—from which expectation values and correlation functions are to be calculated; in general no positive-definite distribution of this form exists. The distribution of time-averaged auxiliary fields is the convolution of this operator distribution with a Gaussian of variance proportional to temperature, and is interpreted as a Boltzmann distribution exp(-βVeff) in classical configuration space. At high temperatures and large degeneracies this classical effective Hamiltonian Veff tends to the static approximation as a classical limit. In the low-temperature limit the field distribution becomes a Wigner function, the sign problem occurs, and Veff is complex. Interpretations of the distributions, and a criterion for their positivity, are discussed. The theory is illustrated by an exact evaluation of the Wigner function for spin s and the effective classical Hamiltonian for the spin-1/2 van der Waals model. The field distribution can be negative here, more noticeably if the number of spins is odd.
CitationSAMSON, J.H., 1995. Classical effective Hamiltonians, Wigner functions, and the sign problem. Physical Review B, 51(1), pp 223–233
Publisher© American Physical Society
NotesThis article was published in the journal, Physical Review B [© American Physical Society]. It is also available at: http://link.aps.org/abstract/PRB/v51/p223.