Exact classical effective potentials

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.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.

arXiv:quant-ph/9905089May 26, 19990 citations

Classical effective Hamiltonians, Wigner functions, and the sign problem

John H. 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(-\ensuremath{\beta} ${\mathit{V}}_{\mathrm{eff}}$ ) in classical configuration space. At high temperatures and large degeneracies this classical effective Hamiltonian ${\mathit{V}}_{\mathrm{eff}}$ 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 ${\mathit{V}}_{\mathrm{eff}}$ 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.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}${\mathit{V}}_{\mathrm{eff}}$) in classical configuration space. At high temperatures and large degeneracies this classical effective Hamiltonian ${\mathit{V}}_{\mathrm{eff}}$ 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 ${\mathit{V}}_{\mathrm{eff}}$ 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.

Physical review. B, Condensed matterJan 1, 19958 citations

The Wigner Quasiprobability and Its Integration over the Radius for Coherent States

Alfred Wünsche

The first part is a systematical representation of the Wigner quasiprobability in representations by canonical variables and pairs of complex conjugate variables with particular attention to the reconstruction of the density operator from the Wigner quasiprobability to show their equivalence. All formulae are derived from basic postulates of quantum mechanics. Problems of operator orderings are touched. The second part considers the integration over the amplitude in the Wigner quasiprobability and its specialization for coherent states and for even and odd coherent states and provides phase distributions which are presented in graphics that are new in their kind. It is demonstrated that the Heitler number phase uncertainty relation is incorrect. The problem of the transition from quantum to classical distribution functions by ℏ→0 is discussed with the conjecture that the inertial point of the distributions plays a basic role.The first part is a systematical representation of the Wigner quasiprobability in representations by canonical variables and pairs of complex conjugate variables with particular attention to the reconstruction of the density operator from the Wigner quasiprobability to show their equivalence. All formulae are derived from basic postulates of quantum mechanics. Problems of operator orderings are touched. The second part considers the integration over the amplitude in the Wigner quasiprobability and its specialization for coherent states and for even and odd coherent states and provides phase distributions which are presented in graphics that are new in their kind. It is demonstrated that the Heitler number phase uncertainty relation is incorrect. The problem of the transition from quantum to classical distribution functions by ℏ→0 is discussed with the conjecture that the inertial point of the distributions plays a basic role.

Journal of Modern PhysicsJan 1, 20260 citations

Weighted Phase-Space Paths for Exact Wigner Dynamics

Surachate Limkumnerd, Panat Phanthaphanitkul

A quantum state can be written in phase space, but the resulting object is not generally the probability density of a positive stochastic process on ordinary phase space. We spell this out for Wigner dynamics. If a positive phase-space process is required only to reproduce the Born density after integrating over momentum, the requirement fixes only an integrated current; the local drift and diffusion remain underdetermined. If one instead requires all Weyl-ordered expectation values, the phase-space object is fixed to be the Wigner function. For non-quadratic potentials the Wigner--Moyal generator contains higher-order, signed momentum-transfer terms, so it is not the Fokker--Planck generator of a positive Brownian diffusion. The exact Wigner function must therefore be reconstructed, in a stochastic representation, as a weighted empirical measure \[ \FW(\z,t)=\E_{\Pp}[W_tδ(\z-\z_t)], \qquad \z=(q,p), \] rather than the unweighted density of sampled carrier trajectories. With classical Hamiltonian flow as the carrier, all nonclassical correction beyond classical transport sits in the Moyal residual and can be represented by signed weights or branching events. The same split defines a residual diagnostic that vanishes for quadratic Hamiltonians and measures what classical carrier transport misses in anharmonic dynamics. The formulation also gives a forward--reverse relation for signed Wigner path measures. The ratio of forward and reversed contributions separates into a positive magnitude factor and a sign factor. This sign records the parity of the Wigner interference contribution; it is not a thermodynamic entropy production.A quantum state can be written in phase space, but the resulting object is not generally the probability density of a positive stochastic process on ordinary phase space. We spell this out for Wigner dynamics. If a positive phase-space process is required only to reproduce the Born density after integrating over momentum, the requirement fixes only an integrated current; the local drift and diffusion remain underdetermined. If one instead requires all Weyl-ordered expectation values, the phase-space object is fixed to be the Wigner function. For non-quadratic potentials the Wigner--Moyal generator contains higher-order, signed momentum-transfer terms, so it is not the Fokker--Planck generator of a positive Brownian diffusion. The exact Wigner function must therefore be reconstructed, in a stochastic representation, as a weighted empirical measure \[ \FW(\z,t)=\E_{\Pp}[W_tδ(\z-\z_t)], \qquad \z=(q,p), \] rather than the unweighted density of sampled carrier trajectories. With classical Hamiltonian flow as the carrier, all nonclassical correction beyond classical transport sits in the Moyal residual and can be represented by signed weights or branching events. The same split defines a residual diagnostic that vanishes for quadratic Hamiltonians and measures what classical carrier transport misses in anharmonic dynamics. The formulation also gives a forward--reverse relation for signed Wigner path measures. The ratio of forward and reversed contributions separates into a positive magnitude factor and a sign factor. This sign records the parity of the Wigner interference contribution; it is not a thermodynamic entropy production.

quant-pharXiv:2605.05764May 8, 20260 citations

Simulation of the time evolution of the Wigner function with a first-principles Monte Carlo method

M. S. Torres, G. Tosi, J. M. A. Figueiredo

The implementation of Monte Carlo methods acting in the quantum phase space is hindered by the fact that quantum phase-space information is available only through quasiprobability densities. In this work, we present a first-principles Monte Carlo method employing a hidden variables representation. This allows the full quantum time evolution of an arbitrary initial quantum state to be calculated by a classical Monte Carlo algorithm, even for systems subjected to time-dependent potentials. Guidelines for implementing a practical algorithm are presented.The implementation of Monte Carlo methods acting in the quantum phase space is hindered by the fact that quantum phase-space information is available only through quasiprobability densities. In this work, we present a first-principles Monte Carlo method employing a hidden variables representation. This allows the full quantum time evolution of an arbitrary initial quantum state to be calculated by a classical Monte Carlo algorithm, even for systems subjected to time-dependent potentials. Guidelines for implementing a practical algorithm are presented.

Physical Review ESep 4, 20092 citations

Quantum Monte Carlo scheme for frustrated Heisenberg antiferromagnets

Jacek Wojtkiewicz

When one tries to simulate quantum spin systems by the Monte Carlo method, often the ``minus-sign problem'' is encountered. In such a case, an application of probabilistic methods is not possible. In this paper, the method has been proposed which allows to avoid the minus-sign problem for certain class of frustrated Heisenberg models. The systems where this method is applicable are, for instance, the pyrochlore lattice and the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model. The method works in singlet sector. It relies on the expression of wave functions in dimer (pseudo)basis and writing down the Hamiltonian as a sum over plaquettes. In such a formulation, matrix elements of the exponent of Hamiltonian are positive.When one tries to simulate quantum spin systems by the Monte Carlo method, often the ``minus-sign problem'' is encountered. In such a case, an application of probabilistic methods is not possible. In this paper, the method has been proposed which allows to avoid the minus-sign problem for certain class of frustrated Heisenberg models. The systems where this method is applicable are, for instance, the pyrochlore lattice and the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model. The method works in singlet sector. It relies on the expression of wave functions in dimer (pseudo)basis and writing down the Hamiltonian as a sum over plaquettes. In such a formulation, matrix elements of the exponent of Hamiltonian are positive.

arXiv:cond-mat/0611550Physical Review BMay 15, 20072 citations

{\mathit{V}}_{\mathrm{eff}}

) in classical configuration space. At high temperatures and large degeneracies this classical effective Hamiltonian

{\mathit{V}}_{\mathrm{eff}}

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

{\mathit{V}}_{\mathrm{eff}}

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.