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Statistical Mechanics and the Physics of the
Many-Particle Model Systems∗

arXiv:1101.3423v1 [cond-mat.str-el] 18 Jan 2011

A. L. Kuzemsky
Bogoliubov Laboratory of Theoretical Physics,
Joint Institute for Nuclear Research,
141980 Dubna, Moscow Region, Russia.˜kuzemsky

The development of methods of quantum statistical mechanics is considered in light of
their applications to quantum solid-state theory. We discuss fundamental problems of the
physics of magnetic materials and the methods of the quantum theory of magnetism, including the method of two-time temperature Green’s functions, which is widely used in various
physical problems of many-particle systems with interaction. Quantum cooperative effects and
quasiparticle dynamics in the basic microscopic models of quantum theory of magnetism: the
Heisenberg model, the Hubbard model, the Anderson Model, and the spin-fermion model are
considered in the framework of novel self-consistent-field approximation. We present a comparative analysis of these models; in particular, we compare their applicability for description
of complex magnetic materials. The concepts of broken symmetry, quantum protectorate,
and quasiaverages are analyzed in the context of quantum theory of magnetism and theory
of superconductivity. The notion of broken symmetry is presented within the nonequilibrium
statistical operator approach developed by D.N. Zubarev. In the framework of the latter approach we discuss the derivation of kinetic equations for a system in a thermal bath. Finally,
the results of investigation of the dynamic behavior of a particle in an environment, taking
into account dissipative effects, are presented.
Keywords: Quantum statistical physics; quantum theory of magnetism; theory of superconductivity; Green’s function method; Hubbard model and other many-particle models on a
lattice; symmetry principles; breaking of symmetries; Bogoliubov’s quasiaverages; quasiparticle many-body dynamics; magnetic polaron; microscopic theory of the antiferromagnetism.
PACS: 05.30.-d, 05.30.Fk, 74.20.-z, 75.10.-b

Physics of Particles and Nuclei, 2009, Vol. 40, No. 7, pp. 949-997.