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Journal of Mathematical Physics

Equilibrium states of inhomogeneous mean-field quantum spin systems with random sites

A. Blobel, Joachim A. Messer

Published May 1, 1985• 2 citations
Publisher

Abstract

For inhomogeneous mean-field quantum spin systems with random sites, we prove that the weak*-limit points of the sequence of n-particle Gibbs states, on a suitable C*-algebra, are superpositions of inhomogeneous product states, which are given by solutions of a generalized Curie–Weiss equation. Under certain conditions there is only one weak*-limit point. As applications, the BCS model and a spin system with cosine interaction are studied.

DOI
10.1063/1.526536

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For inhomogeneous mean-field quantum spin systems with random sites, we prove that the weak*-limit points of the sequence of n-particle Gibbs states, on a suitable C*-algebra, are superpositions of inhomogeneous product states, which are given by solutions of a generalized Curie–Weiss equation. Under certain conditions there is only one weak*-limit point. As applications, the BCS model and a spin system with cosine interaction are studied.
We show the existence of the limiting free-energy density of inhomogeneous ( site-dependent coupling ) mean-field models in the thermodynamic limit, and derive a variational formula for this quantity. The formula requires the minimization of an energy term plus an entropy term as a functional depending on a function with values in the one-particle state space. The minimizing functions describe the pure phases of the system, and all cluster points of the sequence of finite volume equilibrium states have unique integral decomposition into pure phases. Applications are considered; they include the full BCS-model, and random mean-field models
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We study the relation between quantum mechanical and classical Gibbs states of spin systems with spin quantum number sss. It is known that quantum states and observables can be represented by functions defined on the phase space S{\mathcal S}S, which in our case is the NNN-fold product of unit spheres. Therefore, the classical limit s→∞s\to\inftys→∞ of (suitably scaled) quantum Gibbs states can be described as the limit of functions defined on S{\mathcal S}S. We choose to approximate the exponential function of the Hamiltonian by a polynomial of degree nnn and thus have to deal with the problem of the limit of double sequences (depending on nnn and sss) treated in the theorem of Moore-Osgood. The convergence of quantum Gibbs states to classical ones is illustrated by the example of the Heisenberg dimer. We apply our method to the explicit calculation of the phase space function describing spin monomials, and finally add some general remarks on the theory of spin coherent states.
The statistical thermodynamics of the quantum spin- s XXZ Heisenberg model in the Weiss (mean-field) limit rests on the outer multiplicities of the irreducible representations in the Clebsch–Gordan decomposition of the n -fold spin- s tensor product, which determine the energy-level degeneracies and hence the partition function. Using known closed forms for these multiplicities in terms of generalized binomial coefficients of order 2 s + 1 , we write the exact finite-size partition function of the effective collective Hamiltonian and evaluate its thermodynamic limit by a large-deviation (Laplace) analysis validated against direct summations for systems of up to n = 10 4 spins. The free energy then follows from a two-parameter variational principle in the total spin and the magnetization per site. The ferromagnetic ordering is driven by the rotationally invariant part of the Hamiltonian, so that the Curie temperature T c = 1 3 J s ( s + 1 ) is independent of the anisotropy Δ, and the transition is of second order for every spin s , with classical mean-field exponents. The anisotropy reshapes the phase diagram: for Δ < 1 the field-driven first-order line of the isotropic ( Δ = 1 ) model opens into a dome of easy-plane ferromagnetic order, bounded by a continuous transition that restores the residual U(1) symmetry, and no first-order transitions or metastability occur throughout − 1 ≤ Δ < 1 ; the isotropic Curie-Weiss ferromagnet is recovered as Δ → 1. In two appendices we detail the asymptotics of the outer multiplicities down to the Gaussian-fluctuation prefactor and discuss the q -number and Jacobi-theta structures underlying the model, including the partial-theta-type structure that obstructs a standard theta-function reduction of the residual total-spin sum.