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International Journal of Modern Physics B

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

John H. Samson

Published Jan 1, 1993• 1 citation
Publisher

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.

DOI
10.1142/s0217979293001244

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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.
The partition function of interacting electrons is often represented as that of noninteracting electrons moving in a stochastic time-dependent field, integrated over the field. Quantum Monte Carlo computation of this functional integral suffers from the sign problem: rapid oscillation of the sign of the integrand at low temperature. The integrand is in general complex: the phase tends to a Berry phase for smooth paths at low temperatures, where spins follow fields adiabatically, and to zero in the high-temperature limit, recovering the static approximation.
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(-\ensuremath{\beta}Veff{\mathit{V}}_{\mathrm{eff}}Veff​) in classical configuration space. At high temperatures and large degeneracies this classical effective Hamiltonian Veff{\mathit{V}}_{\mathrm{eff}}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{\mathit{V}}_{\mathrm{eff}}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.
The auxiliary-field quantum Monte Carlo method is reviewed. The Hubbard-Stratonovich transformation converts an interacting Hamiltonian into a non-interacting Hamiltonian in a time-dependent stochastic field, allowing calculation of the resulting functional integral by Monte Carlo methods. The method is presented in a sufficiently general form to be applicable to any Hamiltonian with one- and two-body terms, with special reference to the Heisenberg model and one- and many-band Hubbard models. Many physical correlation functions can be related to correlation functions of the auxiliary field; general results are given here. Issues relating to the choice of auxiliary fields are addressed; operator product identities change the relative dimensionalities of the attractive and repulsive parts of the interaction. Frequently the integrand is not positive-definite, rendering numerical evaluation unstable. If the auxiliary field violates time-reversal invariance, the integrand is complex and this sign problem becomes a phase problem. The origin of this sign or phase is examined from a number of geometrical and other viewpoints and illustrated by simple examples: the phase problem by the spin (1/2) Heisenberg model, and the sign problem by the attractive SU(N) Hubbard model on a triangular molecule with negative hopping integrals. In the latter case, widely studied in the Jahn-Teller literature, the sign is due neither to fermions nor spin, but to frustration. This system is used to illustrate a number of suggested interpretations of the sign problem.
A quantum spin system can be modelled by an equivalent classical system, with an effective Hamiltonian obtained by integrating all non-zero frequency modes out of the path integral. The effective Hamiltonian Heff({Si}) derived from the coherent-state integral is highly singular: the quasiprobability density exp(−βHeff), a Wigner function, imposes quantisation through derivatives of delta functions. This quasiprobability is the distribution of the timeaveraged lower symbol of the spin in the coherent-state integral. We relate the quantum Monte Carlo minus-sign problem to the non-positivity of this quasiprobability, both analytically and by Monte Carlo integration. 1
Numerical evaluation of functional integrals usually involves a finite ( L -slice) discretization of the imaginary-time axis. In the auxiliary-field method, the L -slice approximant to the density matrix can be evaluated as a function of inverse temperature at any finite L as L (β) = [ 1 (β/ L )] L , if the density matrix 1 (β) in the static approximation is known. We investigate the convergence of the partition function Z L (β) ≡ Tr L (β), the internal energy and the density of states g L ( E ) (the inverse Laplace transform of Z L ), as L → ∞ . For the simple harmonic oscillator, g L ( E ) is a normalized truncated Fourier series for the exact density of states. When the auxiliary-field approach is applied to spin systems, approximants to the density of states and heat capacity can be negative. Approximants to the density matrix for a spin-½ dimer are found in closed form for all L by appending a self-interaction to the divergent Gaussian integral and analytically continuing to zero self-interaction. Because of this continuation, the coefficient of the singlet projector in the approximate density matrix can be negative. For a spin dimer, Z L is an even function of the coupling constant for L < 3: ferromagnetic and antiferromagnetic coupling can be distinguished only for L ⩾3, where a Berry phase appears in the functional integral. At any non-zero temperature, the exact partition function is recovered as L → ∞ .
A quantum spin system can be modelled by an equivalent classical system, with an effective Hamiltonian obtained by integrating all non-zero frequency modes out of the path integral. The effective Hamiltonian H_eff(S_i) derived from the coherent-state integral is highly singular: the quasiprobability density exp(-beta H_eff), a Wigner function, imposes quantisation through derivatives of delta functions. This quasiprobability is the distribution of the time-averaged lower symbol of the spin in the coherent-state integral. We relate the quantum Monte Carlo minus-sign problem to the non-positivity of this quasiprobability, both analytically and by Monte Carlo integration.