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Journal of Mathematical Physics

Equilibrium states of a class of quantum mean-field theories

Pavel Bóna

Published Dec 1, 1989• 21 citations
Publisher

Abstract

The general structure and existence of equilibrium and ground states of infinite spin systems with the mean-field type interaction of Hepp and Lieb [Helv. Phys. Acta 46, 573 (1973)] is investigated. The form and the role of the ‘‘Bogoliubov–Haag’’ Hamiltonians is cleared. The analysis is based on a recent mathematically correct definition of the corresponding time automorphism group τQ (specified by a classical Hamiltonian Q) of an appropriately defined ‘‘physical’’ C*-algebra of observables 𝒞 containing an abelian subalgebra 𝒩 of classical macroscopic quantities as well as the conventional ‘‘bare’’ quasilocal (sub-) algebra 𝒜, 𝒞=𝒜⊗𝒩 [P. Bóna, J. Math. Phys. 29, 2223 (1988)]. A Lie group G of ‘‘hidden symmetries’’ acting nontrivially (also) on 𝒩 plays an essential role in the presented theory. The applications to the quasispin strong-coupling version of the BCS model of superconductivity, and to the corresponding model of the Josephson junction illustrate possibilities of the developed formalism as well as an easy manipulation with it. A simple mathematical description of ‘‘symmetry breaking’’ is sketched for the considered class of mathematical models of large quantal systems.

DOI
10.1063/1.528489

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The general structure and existence of equilibrium and ground states of infinite spin systems with the mean-field type interaction of Hepp and Lieb [Helv. Phys. Acta 46, 573 (1973)] is investigated. The form and the role of the ‘‘Bogoliubov–Haag’’ Hamiltonians is cleared. The analysis is based on a recent mathematically correct definition of the corresponding time automorphism group τQ (specified by a classical Hamiltonian Q) of an appropriately defined ‘‘physical’’ C*-algebra of observables 𝒞 containing an abelian subalgebra 𝒩 of classical macroscopic quantities as well as the conventional ‘‘bare’’ quasilocal (sub-) algebra 𝒜, 𝒞=𝒜⊗𝒩 [P. Bóna, J. Math. Phys. 29, 2223 (1988)]. A Lie group G of ‘‘hidden symmetries’’ acting nontrivially (also) on 𝒩 plays an essential role in the presented theory. The applications to the quasispin strong-coupling version of the BCS model of superconductivity, and to the corresponding model of the Josephson junction illustrate possibilities of the developed formalism as well as an easy manipulation with it. A simple mathematical description of ‘‘symmetry breaking’’ is sketched for the considered class of mathematical models of large quantal systems.
An infinite quantum system with correctly defined dynamics τQ as an automorphism group of a C*-algebra 𝒞 of observables is determined by any continuous unitary representation U(G) of a connected Lie group G, as well as by an arbitrary differentiable real function Q on the dual space 𝔤* to the Lie algebra 𝔤 of G with the canonically defined Poisson flow φQ on 𝔤*. For specific choices of Q and G, the system can be obtained as the thermodynamic limit of a net of finite lattice systems with the mean-field type interaction of Hepp and Lieb [Helv. Phys. Acta 46, 573 (1973)]. A simple nontrivial model of this type is the quasispin BCS model of superconductivity in the strong coupling limit, or a corresponding model of the Josephson junction. A peculiar feature of the considered models is τQ noninvariance of the usually considered C*-algebra 𝒜 of quasilocal observables, as well as an important role of classical dynamics of a set of macroscopic (intensive) observables Qφ in the description of τQ. The work is restricted here to norm-continuous representations U(G), in which case 𝒞 is isomorphic to the tensor product 𝒜⊗𝒩, where 𝒩 is the commutative algebra of classical (intensive) observables of the considered infinite quantum system.
Following the work of P. Bona [J. Math. Phys. 29, 2223 (1988)], a class of infinite mean-field quantum lattice systems is considered within the framework of algebraic quantum mechanics, and the following question is discussed: Which are the possible dynamical descriptions of the system, i.e., in which representations π of the system’s quasilocal C*-algebra 𝒜 do the local time evolutions converge in the thermodynamic limit to an automorphism group of a certain enlarged C*-algebra 𝒞π containing π(𝒜) and (some of) the classical observables of the system in π. To this end we associate a ‘‘classical state space’’ Eπ to any representation π (which is a subset of the phase-space E⊇RL defined by Bona for the largest possible representation), and show how it is characterized by the π-normal states, thereby obtaining a way to determine Eπ explicitly in many cases. The answer to the question posed then reads: The limiting (Heisenberg) dynamics exists in π on 𝒞π if and only if the set Eπ is invariant under the flow φQt on E which describes the time evolution of classical observables. Conversely, to any invariant, closed subset B of E, there exists a dynamical description of the quantum system with classical state space B.
This is an extended and corrected version of lecture notes originally written for a one semester course at Leibniz University Hannover. The main aim of the notes is to give an introduction to the mathematical methods used in describing discrete quantum systems consisting of infinitely many sites. Such systems can be used, for example, to model the materials in condensed matter physics. The notes provide the necessary background material to access recent literature in the field. Some of these recent results are also discussed. The contents are roughly as follows: (1) quick recap of essentials from functional analysis, (2) introduction to operator algebra, (3) algebraic quantum mechanics, (4) infinite systems (quasilocal algebra), (5) KMS and ground states, (6) Lieb-Robinson bounds, (7) algebraic quantum field theory, (8) superselection sectors of the toric code, (9) Haag-Ruelle scattering theory in spin systems, (10) applications to gapped phases. The level is aimed at students who have at least had some exposure to (functional) analysis and have a certain mathematical maturity.
In the framework of operator algebraic quantum statistical mechanics, the global nonequilibrium dynamics for a general class of interactions of infinite mean field quantum lattice systems with the boson field is investigated. The associated interaction operators consist of arbitrary powers of the collective density operators of the mean field system and are linear with respect to the bosonic field operators. Instead of the usual perturbation expansions, here the interacting dynamics are studied by means of cocycle techniques. The cocycle methods are more appropriate to the considered class of interactions, and lead to explicit closed expressions for the dynamical automorphism groups. The cocycle equations connect the classical, collective dynamical behavior of the mean field system with the one-boson dynamics. In physical applications such systems are due to collectively ordered finite-level atoms or the Josephson junction in the thermodynamic limit weakly interacting with the electromagnetic field.
It is shown how can a given kinematical symmetry group G of an infinite quantum system determine a set of classical-mechanical macroscopic variables with a natural Poisson structure on it. For the long-range interactions of the type introduced by Hepp and Lieb [Helvetica Phys.Acta 46, 573 (1973)] the dynamics of the large quantal system [P.Bona, J.Math.Phys. 29, 2223(1988)] leads to the classical Hamiltonian motion of the macroscopic quantities determined by a Hamilton function specified uniquely by the local quantal Hamiltonians. A model of Josephson junction is a nontrivial integrable example of this class of systems. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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.