Bogoliubov's Quasiaverages, Broken Symmetry and Quantum Statistical Physics

TL;DR

This paper reviews Bogoliubov's quasiaverages method and its role in understanding symmetry breaking, quantum protectorates, and emergent phenomena in quantum statistical physics and many-body systems.

Contribution

It unifies the construction of mean fields and self-energies with source fields and clarifies the role of symmetry breaking and quantum protectorates in complex quantum systems.

Findings

Reanalysis of symmetry breaking in many-body systems.

Discussion of quantum protectorates as signatures of emergent behavior.

Generalized scheme for mean fields and self-energy in quantum models.

Abstract

The development and applications of the method of quasiaverages developed by N. N. Bogoliubov to quantum statistical physics and to quantum solid state theory and, in particular, to quantum theory of magnetism, were analyzed. The problem of finding the ferromagnetic, antiferromagnetic and superconducting symmetry broken solutions of the correlated lattice fermion models was discussed within the irreducible Green functions method. A unified scheme for the construction of generalized mean fields (elastic scattering corrections) and self-energy (inelastic scattering) in terms of the Dyson equation was generalized in order to include the source fields. The interrelation of the Bogoliubov's idea of quasiaverages and the concepts of symmetry breaking and quantum protectorate was discussed briefly in the context of quantum statistical physics. The idea of quantum protectorate reveals the essential difference in the behaviour of the complex many-body systems at the low-energy and high-energy scales. It was shown that the role of symmetry (and the breaking of symmetries) in combination with the degeneracy of the system was reanalyzed and essentially clarified within the framework of the method of quasiaverages. The complementary notion of quantum protectorate might provide distinctive signatures and good criteria for a hierarchy of energy scales and the appropriate emergent behavior.