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arXiv:1607.03024

On ergodic states, spontaneous symmetry breaking and the Bogoliubov quasi-averages

Walter Felipe Wreszinski, Valentin Anatol'evich Zagrebnov

Published Jul 7, 2016• 2 citations
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Abstract

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.

DOI
10.48550/arxiv.1607.03024

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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.
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.
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.
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.
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.
We study Bose–Einstein condensation of a charged relativistic boson system as a phenomenon of spontaneous symmetry breaking. By using the Bogoliubov's quasi-average method, it is found that the vacuum expectation values of the scalar fields are nonvanishing, indicating the nonperturbative nature of the condensation. The vacuum of the condensate is explicitly constructed, which is a superposition of states with different charges. A relation is found between the vacuum and the Stagerstam's coherent states of definite charge.