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arXiv:1512.06986 · Journal of Mathematical Physics

The free energy in a class of quantum spin systems and interchange processes

Jakob E. Björnberg

Published Jul 1, 2016• 17 citations
arXivHTMLPDF

Abstract

We study a class of quantum spin systems in the mean-field setting of the complete graph. For spin S=12, the model is the Heisenberg ferromagnet, and for general spin S∈12N, it has a probabilistic representation as a cycle-weighted interchange process. We determine the free energy and the critical temperature (recovering results by Tóth and by Penrose when S=12). The critical temperature is shown to coincide (as a function of S) with that of the q = 2S + 1 state classical Potts model, and the phase transition is discontinuous when S ≥ 1.

DOI
10.1063/1.4959238

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We study a class of quantum spin systems in the mean-field setting of the complete graph. For spin S=12, the model is the Heisenberg ferromagnet, and for general spin S∈12N, it has a probabilistic representation as a cycle-weighted interchange process. We determine the free energy and the critical temperature (recovering results by Tóth and by Penrose when S=12). The critical temperature is shown to coincide (as a function of S) with that of the q = 2S + 1 state classical Potts model, and the phase transition is discontinuous when S ≥ 1.
We present a detailed analysis of certain quantum spin systems with inhomogeneous (non-random) mean-field interactions. Examples include, but are not limited to, the interchange- and spin singlet projection interactions on complete bipartite graphs. Using two instances of the representation theoretic framework of Schur–Weyl duality, we can explicitly compute the free energy and other thermodynamic limits in the models we consider. This allows us to describe the phase transition, the ground-state phase diagram, and the expected structure of extremal states.
Abstract We study a two-parameter family of quantum spin systems on the complete graph, which is the most general model invariant under the complex orthogonal group. In spin S=12S=\frac {1}{2}S=21​ it is equivalent to the XXZ model, and in spin S=1S=1S=1 to the bilinear-biquadratic Heisenberg model. The paper is motivated by the work of Björnberg, whose model is invariant under the (larger) complex general linear group. In spin S=12S=\frac {1}{2}S=21​ and S=1S=1S=1 we give an explicit formula for the free energy for all values of the two parameters, and for spin S>1S>1S>1 for when one of the parameters is non-negative. This allows us to draw phase diagrams and determine critical temperatures. For spin S=12S=\frac {1}{2}S=21​ and S=1S=1S=1, we give the left and right derivatives as the strength parameter of a certain magnetisation term tends to zero, and we give a formula for a certain total spin observable, and heuristics for the set of extremal Gibbs states in several regions of the phase diagrams, in the style of a recent paper of Björnberg, Fröhlich, and Ueltschi. The key technical tool is expressing the partition function in terms of the irreducible characters of the symmetric group and the Brauer algebra. The parameters considered include, and go beyond, those for which the systems have probabilistic representations as interchange processes.
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.
The purpose of this work is to understand the zero temperature phases and the phase transitions of Heisenberg spin systems which can have an extensive, spontaneous magnetic moment; this entails a study of quantum transitions with an order parameter which is also a non-abelian conserved charge. To this end, we introduce and study a new class of lattice models of quantum rotors. We compute their mean-field phase diagrams and present continuum, quantum field-theoretic descriptions of their low energy properties in different regimes. We argue that, in spatial dimensiond=1, the phase transitions in itinerant Fermi systems are in the same universality class as the corresponding transitions in certain rotor models. We discuss implications of our results for itinerant fermions systems in higherdand for other physical systems.
A variational approach based on Bogoliubov inequality for the free energy is employed in order to treat the quantum spin-1 anisotropic ferromagnetic Heisenberg model in the presence of a crystal field. Within the Bogoliubov scheme an improved pair approximation has been used. The temperature-dependent thermodynamic functions have been obtained and provide much better results than the previous simple mean-field scheme. In one dimension, which is still nonintegrable for quantum spin-1, we get the exact results in the classical limit, or near-exact results in the quantum case, for the free energy, magnetization, and quadrupole moment, as well for the transition temperature. In two and three dimensions the corresponding global phase diagrams have been obtained as a function of the parameters of the Hamiltonian. First-order transition lines, second-order transition lines, tricritical and tetracritical points, and critical endpoints have been located through the analysis of the minimum of the Helmholtz free energy and a Landau-like expansion in the approximated free energy. Only first-order quantum transitions have been found at zero temperature. Limiting cases, such as isotropic Heisenberg, Blume-Capel, and Ising models, have been analyzed and compared to previous results obtained from other analytical approaches as well as from Monte Carlo simulations.