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Physical Review C

Coherent states and the calculation of nuclear partition functions

Osvaldo Civitarese, M. Reboiro, S. Jesgarz, Peter Oto Hess

Published Oct 23, 2001• 1 citation
Publisher

Abstract

Coherent states are introduced as test functions to formulate the statistical mechanics of fermions and bosons interacting via schematic forces. Finite temperature solutions to the Lipkin model and to the Sch\"utte--Da Providencia model are obtained by performing the statistical sum $\`a\mathrm{la}$ Hecht, e.g., by using coherent states. Comparison between present and exacts results is discussed.

DOI
10.1103/physrevc.64.054317

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Coherent states are introduced as test functions to formulate the statistical mechanics of fermions and bosons interacting via schematic forces. Finite temperature solutions to the Lipkin model and to the Sch"utte--Da Providencia model are obtained by performing the statistical sum aˋla\`a\mathrm{la}aˋla Hecht, e.g., by using coherent states. Comparison between present and exacts results is discussed.
Formal functional integrals are commonly used as theoretical tools and as sources of intuition for predicting phase transitions of many-body systems in Condensed Matter Physics. In this thesis, we derive rigorous versions of these functional integrals for two types of quantum many-particle systems. We begin with a brief review of quantum statistical mechanics in Chapter 2 and the formalism of coherent states in Chapter 3, which form the basis for our analysis in Chapters 4 and 5. In Chapter 4, we study a mixed gas of bosons and/or fermions interacting on a finite lattice, with a general Hamiltonian that preserves the total number of particles in each species. We rigorously derive a functional integral representation for the partition function, which employs a large-field cutoff for the boson fields. We then expand the resulting “action” in powers of the fields and find a recursion relation for the coefficients. In the case of a two-body interaction (such as the Coulomb interaction), we also find bounds on the coefficients, which give a domain of analyticity for the action. This domain is large enough for use of the action in the functional integral, provided that the large-field cutoffs are taken to grow not too quickly. In Chapter 5, we study a system of electrons and phonons interacting in a finite lattice, using the Holstein Hamiltonian. Again, we rigorously derive a coherent-state functional integral representation for the partition function of this system and then prove that the “action” in the functional integral is an entire-analytic function of the fields. However, since the Holstein Hamiltonian does not preserve the total number of bosons, the approach from Chapter 4 requires some modification. In particular, we repeatedly use Duhamel expansions in powers of the interaction, rather than sums over particle numbers.
A general method of calculation is described for quantum statistical mechanics. It is based on a simplification of the Laplace transform of the density matrix which follows from a theorem due to Hugenholtz. The basic result is that an element of the density matrix can be written as a sum over graphs, with the contribution of each graph factored into contributions from connected or linked graphs. Applied to the grand partition function, the exponential formula of Bloch and DeDominicis is obtained in a simple way. A similar formula is then derived for the canonical ensemble for the case of a nondegenerate gas. In this way the familiar result of Uhlenbeck and Beth is obtained for the second virial coefficient. Techniques are also introduced for evaluating ensemble averages of operators. In this connection, some care must be exercised in the case of diagonal operators. Finally, these methods are used to calculate the pair-correlation function for a system of fermions interacting through short-range forces.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed