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Physical Review A

Path-integral approaches to the statistical mechanics of quantum systems: Variational methods for influence functionals

Kurt Allinger

Published Jan 1, 1989• 3 citations
Publisher

Abstract

A powerful numerical procedure is developed for calculating temperature-dependent quantities, such as partition functions for quantum systems coupled to thermal or other baths. A variational procedure is worked out to construct model Hamiltonians starting from the general results of the previous paper for influence functionals. Explicit, though intricate, formulas for all relevant quantities are presented. The methods developed are applied to a generalized model for a boson-bath interaction.

DOI
10.1103/physreva.39.881

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A powerful numerical procedure is developed for calculating temperature-dependent quantities, such as partition functions for quantum systems coupled to thermal or other baths. A variational procedure is worked out to construct model Hamiltonians starting from the general results of the previous paper for influence functionals. Explicit, though intricate, formulas for all relevant quantities are presented. The methods developed are applied to a generalized model for a boson-bath interaction.
Feynman path integrals offer a convenient and elegant tool for dealing with the statistical physics of quantum-mechanical systems. By using the Trotter product formula, one can evaluate directly the propagation of many-body quantum systems in imaginary time. The idea of an influence functional arises when one wishes to consider division of the physical system into a primary system and a bath. We present a general approach to the problem of constructing influence functionals, one that is capable of dealing both with Boltzmann baths (classical oscillators) and quantum baths of bosons or fermions. The fermion bath of special interest is generally a submanifold of the electronic state such as the bridge structure connecting chromophores or electron localization subunits which are common in problems of mixed valency and superexchange. The fermion bath is treated by a general rewriting of the influence phase S as a sum of an eigenvalue part SEV{S}_{\mathrm{EV}}SEV​ and a remainder, S0{S}^{0}S0, that describes transitions. Reduction of the original Hamiltonian leads to a reduced Hamiltonian with effective off-diagonal matrix elements and influence functionals which can include memory effects. We present a general formalism for construction of influence functionals, and discuss particular applications to systems of interest, especially in electronic-structure problems. The following paper presents a variational technique for finding effective Hamiltonians.
The path-integral method is used for determination of the quantum corrections to the free energy of nonlinear systems. All quantum effects of the harmonic part of the potential are considered and a variational principle is used to account for the quantum corrections due to the anharmonic part. Correct renormalized frequencies are obtained at any temperature and an effective potential to be inserted in the configurational integral is found. A new general expression for the partition function at any temperature in the low-coupling limit is obtained.
Quantum statistical mechanics has developed primarily through two approaches, pioneered by Gibbs and Feynman, respectively. In Gibbs' method one calculates partition functions from phase-space integrations or sums over stationary states. Alternatively, in Feynman's approach, the focus is on the path-integral formulation. The Hubbard-Stratonovich transformation leads to a functional-integral formulation for calculating partition functions. We outline here the functional integral approach to quantum statistical mechanics, including generalizations and improvements to Hubbard's formulation. We show how the dimensionality of the integrals is reduced exactly, how the problem of assuming an unknown canonical transformation is avoided, how the reality of the partition function in the complex representation is guaranteed, and how the extremum conditions are simplified. This formulation can be applied to general systems, including superconductors.
We consider the quantum paths (in Euclidean time) through a space of N multiparticle states, and we derive the influence functional for a subset of m primary states in this space. The influence functional results from integrating out the occupation of the N-m nonprimary or bath states holding fixed the time-dependent occupation path through the primary states. The division into bath states and primary states is completely general in principle, though in practice, physical considerations will guide the choice of partitioning. The exact influence functional is nonlocal in time and very complicated in form. We argue, however, by a variational calculation that a simplified approximate functional can be constructed which is very accurate. Unlike the standard Gaussian influence functionals which are inapplicable for processes involving the large amplitude fluctuations of charge transfer and fermionic exchange, the influence functionals described herein are appropriate for models of these processes.
Abstract We develop a non-perturbative method for calculating partition functions of strongly coupled quantum mechanical systems with interactions between subsystems described by a path integral of a dual system. The dual path integral is derived starting from non-interacting subsystems at zeroth order and then by introducing couplings of increasing complexity at each order of an iterative procedure. These orders of interactions play the role of a dual time and the full quantum partition function is expressed as a transition amplitude in the dual system. More precisely, it is expressed as a path integral from a deformation-operators dependent initial state at zero time/order to the inverse-temperature dependent final state at later time/order. We provide examples of strongly coupled systems with up to first-order interactions (e.g. Ising model) and arbitrary high-order interactions (e.g. 1+1D1+1\hbox {D}1+1D 1 + 1 D QFT). We also discuss a possible emergence of space-time, quantum field theories and general relativity in context of the dual path integral.
For a system of bosons that interact through a class of general memory kernels, a recurrence relation for the partition function is derived within the path-integral formalism. This approach provides a generalization to previously known treatments in the literature of harmonically coupled systems of identical particles. As an example the result is applied to the specific heat of a simplified model of an open quantum system of bosons, harmonically coupled to a reservoir of distinguishable fictitious masses.