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On the cumulant expansion peculiarity for partition function functional of the cluster de Gennes model

N. A. Korynevskii

Published Jan 1, 2000• 0 citations
Publisher

Abstract

The problem of the functional representation for cluster de Gennes model \npartition function within the collective variables method is discussed. Contrary to the usual Ising model case the coefficients of the obtained partition function functional (cluster cumulants) depend on temperature T and \ntransverse field Γ . Therefore there exists a rigorous limitation on the value \nof Γ parameter at low temperatures. The equation for maximum value Γl , \ntemperature and short-range intracluster interaction V is obtained. The \nsolutions of this equation have been found.

DOI
10.5488/cmp.3.4.737

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The problem of the functional representation for cluster de Gennes model \npartition function within the collective variables method is discussed. Contrary to the usual Ising model case the coefficients of the obtained partition function functional (cluster cumulants) depend on temperature T and \ntransverse field Γ . Therefore there exists a rigorous limitation on the value \nof Γ parameter at low temperatures. The equation for maximum value Γl , \ntemperature and short-range intracluster interaction V is obtained. The \nsolutions of this equation have been found.
The problem of the functional representation for cluster de Gennes model partition function within the collective variables method is discussed.Contrary to the usual Ising model case the coefficients of the obtained partition function functional (cluster cumulants) depend on temperature T and transverse field Γ .Therefore there exists a rigorous limitation on the value of Γ parameter at low temperatures.The equation for maximum value Γ l , temperature and short-range intracluster interaction V is obtained.The solutions of this equation have been found.
It is shown for the Ising model that the linked-cluster expansion (LCE) is a special case of the variational cumulant expansion (VCE) in the high-temperature limit. The VCE was originally developed and formulated to treat lattice-gauge-field models. The free energy calculated from the VCE with a trial Hamiltonian of mean-field form in the high-temperature limit is in fact identical to that from the LCE. Thus, a more general expansion that is valid for the whole temperature range is established for the treatment of Ising systems.
The problem of N interacting spins on a lattice is equivalent to one of N clusters linked in a specific manner. The energy of any configuration of spins can be expressed in terms of the energy levels of this cluster. A new expression is obtained for the probability of occurrence of any configuration. A closed form expression is obtained for the partition function per spin in terms of the energy levels of this cluster with the degeneracies being a function of temperature. On physical grounds it is suggested to be the form of the exact partition function per spin. The partition functions of Ising-like systems all have a common form. For the 3D Ising model seven functions need to be determined to describe the partition function completely. The key to understanding second order phase transitions and critical phenomena lies in the temperature dependence of various degeneracies. It is necessary to develop new techniques to determine the partition function that account for this temperature dependence, as it would represent the underlying physics correctly.
We present in this paper a systematic nonperturbative cluster-cumulant method for deriving thermal averages of operators in quantum many-body systems. The method combines the advantages of the cumulant expansion scheme of thermodynamic perturbation theory, the approach of thermofield dynamics as a finite-temperature field theory, and the time-dependent coupled-cluster theory extended to ``imaginary time.'' We have generalized the concepts of cumulants in a nonperturbative manner and have posited on the statistical operator an exponential-like ansatz containing connected, size-extensive operators in the exponent. These latter cumulantlike operators have been termed ``cluster cumulants'' by us. For a compact treatment, we have derived an alternative thermal field theory in which a time-ordered product is expanded in terms of ``thermal normal products'' of operators and thermal contractions---leading to a ``thermal Wick's theorem.'' The thermal normal products are the finite-temperature analogs of the ordinary normal products and have zero thermal averages. Operators in these products commute (anticommute) under permutations for bosons (fermions). This thermal representation is shown to be unitarily related to the traditional thermofield dynamics formulation, but has the advantage of using only the physical variables. The imaginary-time evolution of the statistical operator is treated by our recently formulated time-dependent cluster-cumulant theory. The partition function is evaluated as an exponential of a connected quantity. As an illustrative example, we have computed the partition function of an anharmonic oscillator with equally weighted cubic and quartic perturbation for a wide range of coupling, extending to the strongly nonperturbative regime. We study the behavior of free energy in the low-temperature limit and verify numerically the validity of the Kohn-Luttinger theorem [Phys. Rev. 118, 41 (1960)] for this system. We also show that our formalism is a natural nonperturbative analog of the thermodynamic perturbative theory by showing that a perturbative solution of the thermal-cluster-cumulant equations generates a variation of the Bloch--Balian--de Dominicis theory.
The cluster variation method, which was originally devised for quantum-mechanical many-body systems at zero temperature, is extended to treat finite-temperature systems. The new method is based on a variational principle that the partition function is the maximum of a certain functional, APF (approximate partition function). This method includes the conventional cluster variation method at zero temperature as a limiting case. A procedure is given for calculating APF by cluster expansion for fermion systems. In this procedure, use is made of single-particle energies which should be determined self-consistently. The non-orthogonality problem which comes from the presence of correlations among particles is solved by the use of linear combinations of (generalized) Jastrow wave functions. Further development of this method is briefly discussed.