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arXiv:1704.00190 · Theoretical and Mathematical Physics

Bogoliubov Quasiaverages: Spontaneous Symmetry Breaking and the Algebra of Fluctuations

W. F. Wreszinski, Valentin Anatol'evich Zagrebnov

Published Feb 1, 2018• 12 citations
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Abstract

We present arguments supporting the use of the Bogoliubov method of quasiaverages for quantum systems. First, we elucidate how it can be used to study phase transitions with spontaneous symmetry breaking (SSB). For this, we consider the example of Bose–Einstein condensation in continuous systems. Analysis of different types of generalized condensations shows that the only physically reliable quantities are those defined by Bogoliubov quasiaverages. In this connection, we also solve the Lieb–Seiringer–Yngvason problem. Second, using the scaled Bogoliubov method of quasiaverages and considering the example of a structural quantum phase transition, we examine a relation between SSB and critical quantum fluctuations. We show that the quasiaverages again provide a tool suitable for describing the algebra of critical quantum fluctuation operators in both the commutative and noncommutative cases.

DOI
10.1134/s0040577918020010

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We present arguments supporting the use of the Bogoliubov method of quasiaverages for quantum systems. First, we elucidate how it can be used to study phase transitions with spontaneous symmetry breaking (SSB). For this, we consider the example of Bose–Einstein condensation in continuous systems. Analysis of different types of generalized condensations shows that the only physically reliable quantities are those defined by Bogoliubov quasiaverages. In this connection, we also solve the Lieb–Seiringer–Yngvason problem. Second, using the scaled Bogoliubov method of quasiaverages and considering the example of a structural quantum phase transition, we examine a relation between SSB and critical quantum fluctuations. We show that the quasiaverages again provide a tool suitable for describing the algebra of critical quantum fluctuation operators in both the commutative and noncommutative cases.
It is shown that Bogoliubov quasi-averages select the pure or ergodic states in the ergodic decomposition of the thermal (Gibbs) state. Our examples include quantum spin systems and many-body boson systems. As a consequence, we elucidate the problem of equivalence between Bose-Einstein condensation and the quasi-average spontaneous symmetry breaking (SSB) discussed for continuous boson systems. The multi-mode extended van den Berg-Lewis-Pulé condensation of type III demonstrates that the only physically reliable quantities are those that defined by Bogoliubov quasi-averages.
Abstract Due to high current interest to Bose–Einstein condensation (BEC), the related topics, such as spontaneous gauge symmetry breaking, ergodic decomposition, and particle fluctuations, are intensively discussed in literature. These discussions, unfortunately, involve quite a number of controversies and confusions. The goal of the present brief survey is to clarify some of these confusions, concentrating on the principal points, such as the relationship between the conditions of BEC, the Bogolubov method of quasiaverages, spontaneous gauge symmetry breaking, ergodic decomposition, the presumed existence of nonthermodynamic particle fluctuations leading to the so-called ‘grand canonical catastrophe’, and the requirements for system stability.
In the present interdisciplinary review, we focus on the applications of the symmetry principles to quantum and statistical physics in connection with some other branches of science. The profound and innovative idea of quasiaverages formulated by N. N. Bogoliubov, gives the so-called macro-objectivation of the degeneracy in the domain of quantum statistical mechanics, quantum field theory and quantum physics in general. We discuss the complementary unifying ideas of modern physics, namely: spontaneous symmetry breaking, quantum protectorate and emergence. The interrelation of the concepts of symmetry breaking, quasiaverages and quantum protectorate was analyzed in the context of quantum theory and statistical physics. The chief purposes of this paper were to demonstrate the connection and interrelation of these conceptual advances of the many-body physics and to try to show explicitly that those concepts, though different in details, have certain common features. Several problems in the field of statistical physics of complex materials and systems (e.g., the chirality of molecules) and the foundations of the microscopic theory of magnetism and superconductivity were discussed in relation to these ideas.
Due to high current interest to Bose-Einstein condensation, the related topics, such as spontaneous gauge symmetry breaking, ergodic decomposition, and particle fluctuations, are intensively discussed in literature. These discussions, unfortunately, involve quite a number of controversies and confusions. The goal of the present brief survey is to clarify some of these confusions, concentrating on the principal points, such as the relationship between the conditions of Bose-Einstein condensation, the Bogolubov method of quasiaverages, spontaneous gauge symmetry breaking, ergodic decomposition, the presumed existence of nonthermodynamic particle fluctuations leading to the so-called ``grand canonical catastrophe", and the requirements for system stability.
The realisation of Bose-Einstein condensation under grand-canonical conditions has provided the experimental evidence for the simultaneous occurrence of macroscopic fluctuations and phase coherence of the condensate. The observation of these two features, against a consolidated tradition which wants the fluctuations to be pathological (grand-canonical catastrophe) and incompatible with spontaneous symmetry braking, calls for a comprehensive rethinking of the approach to the problem. In this paper we consider the uniform ideal gas in a box and we present an alternative conceptual framework. We show that the usually-employed Bogoliubov quasi-average construction fails to reproduce the broken-symmetry state. The observed features are accounted for by a different pattern of spontaneous symmetry breaking, characterised by condensation of fluctuations and long-range correlations of the order parameter.
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.