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Classical effective Hamiltonians, Wigner functions, and the sign problem

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journal contribution
posted on 27.04.2006, 13:14 by John SamsonJohn 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.



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SAMSON, J.H., 1995. Classical effective Hamiltonians, Wigner functions, and the sign problem. Physical Review B, 51(1), pp 223–233


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This 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.





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