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Zeitschrift für Naturforschung A

The Quantum Statistical Free Energy Minimum Principle for Multi-Lattice Mean Field Theories

Th. Gerisch, Alfred Rieckers

Published Aug 1, 1990• 8 citations
Publisher

Abstract

Abstract The quantum statistical frame for infinite multi-lattice spin systems is introduced. The thermodynamic functionals specific internal energy, entropy and free energy are shown to exist on the set of permutation invariant states for polynomial mean field interactions by direct estimation methods. Their dependence on the relative sizes of the sublattice systems is made explicit. The set of homogeneous minimal free energy states is shown to be a Bauer simplex which contains all limiting Gibbs states. For the extremal minimal free energy states the self-consistency equations are derived.

DOI
10.1515/zna-1990-0801

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Abstract The quantum statistical frame for infinite multi-lattice spin systems is introduced. The thermodynamic functionals specific internal energy, entropy and free energy are shown to exist on the set of permutation invariant states for polynomial mean field interactions by direct estimation methods. Their dependence on the relative sizes of the sublattice systems is made explicit. The set of homogeneous minimal free energy states is shown to be a Bauer simplex which contains all limiting Gibbs states. For the extremal minimal free energy states the self-consistency equations are derived.
The general structure of thermodynamic equilibrium states for a class of quantum mechanical (multi-lattice) systems is elaborated, combining quantum statistical and thermodynamical methods. The quantum statistical formulation is performed in terms of recent operator algebraic concepts emphasizing the role of the permutation symmetry due to homogeneous coarse graining and employing the internal symmetries. The variational principle of the free energy functional is derived, which determines together with the symmetries the general form of the limiting Gibbs states in terms of their central decomposition. The limiting minimal free energy density and its possible equilibrium states are analyzed on various levels of the description by means of convex analysis, where the Fenchel transforms of the free energies provide entropy like potentials. On the thermodynamic level a modified entropy surface is obtained, which specifies only in combination with its concave envelope the regions of pure and mixed phase states. The symmetry properties of a certain model allow to specify the (non-) differentiability of the minimal free energy density. A characterization and classification of phase transitions in terms of quantum statistical equilibrium states is proposed, and the connection to the Landau theory is established demonstrating that the latter implies a (continuous) deformation of the sets of equilibrium states along a canonically given curve.
We extend Araki's well-known results on the equivalence of the KMS condition and the variational principle for equilibrium states of quantum lattice systems with short-range interactions, to a large class of models possibly containing mean-field interactions (representing an extreme form of long-range interactions). This result is reminiscent of van Hemmen's work on equilibrium states for mean-field models. The extension was made possible by our recent outcomes on states minimizing the free energy density of mean-field models on the lattice, as well as on the infinite volume dynamics for such models.
Quantum many-body states that frequently appear in physics often obey an entropy scaling law, meaning that an entanglement entropy of a subsystem can be expressed as a sum of terms that scale linearly with its volume and area, plus a correction term that is independent of its size.We conjecture that these states have an efficient dual description in terms of a set of marginal density matrices on bounded regions, obeying the same entropy scaling law locally.We prove a restricted version of this conjecture for translationally invariant systems in two spatial dimensions.Specifically, we prove that a translationally invariant marginal obeying three nonlinear constraints-all of which follow from the entropy scaling law straightforwardlymust be consistent with some global state on an infinite lattice.Moreover, we derive a closed-form expression for the maximum-entropy density compatible with those marginals, deriving a variational upper bound on the thermodynamic free energy.The main assumptions of our construction are satisfied exactly by solvable models of topological order and approximately by finite-temperature Gibbs states of certain quantum spin Hamiltonians.
We consider two quantum spin models on kkk-partite graphs. The system is assumed to be invariant under all permutations within any of the kkk sets. In the limit of infinitely many particles, the Gibbs state is described as a mixture of independent spins due to a version of the Quantum de Finetti theorem. First, we study a spin-1/2 antiferromagnetic Heisenberg model and identify its three phases. At low temperature all kkk sets are magnetised but cancel each other, at medium temperature there is a net magnetic field and at high temperature there is no magnetisation. The magnetisation temperature is a solution to a kkk-degree polynomial. Second, we allow a general spin sss and identify all orthogonally invariant models which have frustration. The criterium for a frustration-free model is that the kkk-sets can be partitioned into two parts, where the only antiferromagnetic interaction is between the two parts.
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
The general formalism which connects quantum statistical and thermodynamical phase structures, developed previously, is applied here to a class of FCC-multi-lattice systems with four sublattices. A careful treatment of the symmetries gives insights into the origins of the rich phase structure. Arguments for the existence of the limiting Gibbs states are provided. A microscopic foundation of the Landau scenario for phase transitions, which predicts the active directions and the order parameters, is formulated for the considered models.