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American Journal of Physics

Cumulant expansions in statistical and quantum physics

H. A. Gersch

Published Mar 1, 1991• 1 citation
Publisher

Abstract

Two examples are given of the use of cumulant expansions for systems of particles with hard-core two-body interactions. One of these, in classical statistical mechanics, is shown to lead to virial coefficients. An analogous procedure in quantum mechanics produces energy level shifts due to hard-core interactions that are expressed in terms of the t matrix.

DOI
10.1119/1.16576

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Two examples are given of the use of cumulant expansions for systems of particles with hard-core two-body interactions. One of these, in classical statistical mechanics, is shown to lead to virial coefficients. An analogous procedure in quantum mechanics produces energy level shifts due to hard-core interactions that are expressed in terms of the t matrix.
An expansion for quantum statistical mechanics is derived that gives classical statistical mechanics as the leading term. Each quantum correction comes from successively larger permutation loops, which arise from the factorization of the symmetrization of the wave function with respect to localized particle interchange. Explicit application of the theory yields the full fugacity expansion for the quantum ideal gas, and the second fugacity coefficient for interacting quantum particles, which agree with known results. Compared to the Lee-Yang virial cluster expansion, the present expansion is expected to be more rapidly converging and the individual terms appear to be simpler to evaluate. The results obtained in this paper are intended for practical computer simulation algorithms for terrestrial condensed matter quantum systems.
(received; accepted) PACS. 05.30.-d – Quantum statistical mechanics. PACS. 75.10.Jm – Quantized spin models. PACS. 75.10.Hk – Classical spin models. Abstract. – A method for obtaining a systematic expansion of thermodynamic functions of spin systems with large spin S in powers of 1/S is proposed. This method uses the cumulant technique and a coherent-state representation of the partition function Z. The expansion of Z in terms of cumulants yields an effective classical Hamiltonian with temperature-dependent quantum corrections. For the Heisenberg quantum Hamiltonian, they have a non-Heisenberg form. The effective Hamiltonian can be solved by methods familiar for classical systems. In recent years considerable interest in condensed matter physics has been focusing on the properties of new magnetic materials, such as magnetic molecules with a large effective spin S, or intermetallic compounds containing magnetic layers or chains. For the latter, most of the theoretical studies deal with S = 1/2 systems which, in the one-dimensional case, allow for analytical solutions using special methods such as the Bethe Ansatz or Jordan-Wigner
In the probabilistic approach to quantum many-body systems, the ground-state energy is the solution of a nonlinear scalar equation written either as a cumulant expansion or as an expectation with respect to a probability distribution of the potential and hopping (amplitude and phase) values recorded during an infinitely lengthy evolution. We introduce a perturbative expansion of this probability distribution which conserves, at any order, a multinomial-like structure, typical of uncorrelated systems, but includes, order by order, the statistical correlations provided by the cumulant expansion. The proposed perturbative scheme is successfully tested in the case of pseudo spin 1/2 hard-core boson Hubbard models also when affected by a phase problem due to an applied magnetic field.
Cumulants represent a natural language for expressing macroscopic properties of many-body systems. The most important property of cumulants is that of size consistency, i.e. a cumulant expression for an extensive variable scales with the size of the system, independent of possible further approximations used in the evaluation procedure. Cumulants can be considered as a generalization of linked diagrams known from diagrammatic technique of many-body theory. In this paper we outline a recently introduced method based on cumulants in order to derive expressions for zero-temperature properties of many-particle systems, i.e. the ground-state energy, static expectation values and dynamical correlation functions. This cumulant formalism allows one to describe weakly and strongly correlated systems along the same lines. We show that the coupled-cluster method known from quantum chemistry can be derived from our cumulant approach. Finally, we demonstrate the usefulness of the cumulant method by applying it to examples from solid-state physics and quantum chemistry.
It is shown that statistical mechanics is applicable to quantum systems with finite numbers of particles, such as complex atoms, atomic clusters, etc., where the residual two-body interaction is sufficiently strong. This interaction mixes the unperturbed shell-model basis states and produces ``chaotic'' many-body eigenstates. As a result, an interaction-induced equilibrium emerges in the system, and temperature can be introduced. However, the interaction between the particles and their finite number can lead to prominent deviations of the equilibrium occupation numbers distribution from the Fermi-Dirac shape. For example, this takes place in the cerium atom with four valence electrons, which was used to compare the theory with realistic numerical calculations.