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Journal of Physics A Mathematical and General

High-order perturbation expansion: application to a quantum thermodynamical system of fermions

G. Calucci

Published Nov 1, 1979• 0 citations
Publisher

Abstract

The recently proposed method for investigating the large-order behaviour of the perturbative expansion is applied to a quantum thermodynamical system of interacting fermions. The investigation is performed in detail for a system in one space dimension, with the use of the Thomas-Fermi approximation in order to deal with many-fermion systems. Some observations about the influence of the number of dimensions on the large-order behaviour are presented.

DOI
10.1088/0305-4470/12/11/015

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The recently proposed method for investigating the large-order behaviour of the perturbative expansion is applied to a quantum thermodynamical system of interacting fermions. The investigation is performed in detail for a system in one space dimension, with the use of the Thomas-Fermi approximation in order to deal with many-fermion systems. Some observations about the influence of the number of dimensions on the large-order behaviour are presented.
The recently proposed method for investigating the large-order behaviour of the perturbative expansion is applied to a quantum thermodynamical system of interacting bosons in one space dimension. The method makes essential use of the classical solutions for the total Hamiltonian and of the oscillations about them. The contribution of these oscillations is calculated by using a canonical transformation related to the inverse scattering method, the relevant equation being the nonlinear Schrodinger equation. The leading-order corrections to the Bose-Einstein distribution are obtained explicitly.
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Confined quantum systems involving NNN identical interacting fermions are found in many areas of physics, including condensed matter, atomic, nuclear, and chemical physics. In a previous series of papers, a many-body perturbation method that is applicable to both weakly and strongly interacting systems of bosons has been set forth by the author and coworkers. A symmetry-invariant perturbation theory was developed that uses group theory coupled with the dimension of space as the perturbation parameter to obtain an analytic correlated wave function through first order for a system under spherical confinement with a general two-body interaction. In the present paper, we extend this formalism to large systems of fermions, circumventing the numerical demands of applying the Pauli principle by enforcing the Pauli principle on paper. The method does not scale in complexity with NNN and has minimal numerical cost. We apply the method to a unitary Fermi gas and compare to recent Monte Carlo values.
We review recent advances in the calculations of high-order convergent expansions for quantum many-particle systems. Calculations for ground state properties, including correlation functions and static susceptibilities, for spin models as well as for models of many fermions, such as Hubbard and Kondo models, are discussed. A historical perspective to the subject is provided. Recently important technical advances have been made in perturbative calculations of the excitation spectra of quantum many-particle systems, which enable the calculation of these spectra to high orders. The method, along with its applications, are explained. Fairly comprehensive, though simplified, algorithms for generating lists of relevant clusters, their lattice embeddings and subclusters are presented. The perturbative recursion relations and their computer implementation are also discussed in detail. A compilation is made of various series expansion studies that have been carried out for condensed matter problems. The scope and limitations of these methods are explained, and several open problems are noted.
Abstract We present a new algorithm to evaluate the grand potential at high and finite temperatures using many-body perturbation theory. This algorithm enables us to calculate the contribution of any Hugenholtz or Feynman vacuum diagrams and formulate the results as a sum of divided differences. Additionally, the proposed method is applicable to any interaction in any dimension, allowing us to calculate thermodynamic quantities efficiently at any given temperature, particularly at high temperatures.Furthermore, we apply this algorithm to the Heisenberg spin-1/2 XXZ chain. We obtain all coefficients of the high-temperature expansion of the free energy and susceptibility per site of this model up to the sixth order.
Confined quantum systems involving NNN identical interacting fermions are found in many areas of physics, including condensed matter, atomic, nuclear and chemical physics. In a previous series of papers, a manybody perturbation method that is applicable to both weakly and strongly-interacting systems of bosons has been set forth by the author and coworkers. A symmetry invariant perturbation theory was developed which uses group theory coupled with the dimension of space as the perturbation parameter to obtain an analytic correlated wave function through first order for a system under spherical confinement with a general two-body interaction. In the present paper, we extend this formalism to large systems of fermions, circumventing the numerical demands of applying the Pauli principle by enforcing the Pauli principle on paper. The method does not scale in complexity with NNN and has minimal numerical cost. We apply the method to a unitary Fermi gas and compare to recent Monte Carlo values.