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Long-range Order, Symmetry Breaking, and Order Parameter

Лев П. Питаевский, S. Stringari

Published Jan 1, 2016• 0 citations
Publisher

Abstract

Abstract This chapter introduces the fundamental concepts of off-diagonal long-range order, breaking of gauge symmetry, and order parameter, which underlie the concept of Bose–Einstein condensation. The consequences for the momentum distribution and the possibility of separating a classical term in the Bose field operator associated with the condensate, from a non-condensed component, accounting for thermal and quantum fluctuations, are discussed. The macroscopic classical term provides a natural definition of the complex order parameter, characterized by a modulus and a phase.

DOI
10.1093/acprof:oso/9780198758884.003.0002

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Abstract This chapter introduces the fundamental concepts of off-diagonal long-range order, breaking of gauge symmetry, and order parameter, which underlie the concept of Bose–Einstein condensation. The consequences for the momentum distribution and the possibility of separating a classical term in the Bose field operator associated with the condensate, from a non-condensed component, accounting for thermal and quantum fluctuations, are discussed. The macroscopic classical term provides a natural definition of the complex order parameter, characterized by a modulus and a phase.
The theory of non-interacting Bose gases is supplemented by a numerical quantum field description with a two-dimensional non-local order parameter that allows the modeling of wave-like atomic correlations and interference effects in the limit of low atomic densities. From the present model, it is possible to explain symmetry aspects of non-interacting and very weakly interacting Bose gases in the limit of fluctuating particle numbers, like the forward propagation of time and the relation to the breaking and preservation of phase gauge symmetry in solids. In the present formalism, the propagation of one-directional time arises from the pre-defined and equivalent convergence of independent quantum fields towards the Boltzmann equilibrium, and it is shown that Glauber coherent states are related to the definition of the quantized field. Coherently coupling condensate and non-condensate parts as a direct consequence of the increasing quantum coherence time between the different quantum field components in the Bose gas from cooling to below the critical temperature, the present model describes symmetry breaking, which is originally known from the definition of a specific gauge field from Elitzur's theorem for local gauge fields, as a global physical rather than a purely formal mathematical process.
Spontaneous symmetry breaking originats in quantum mechanical measurement of the relevant observable defining the physical situation, order parameter is the average of this observable. A modification is made on the random-phase postulate validating the ensemble description. Off-diagonal long-range order, macroscopic wavefunction and interference effects in many-particle systems present when there is a so-called nucleation of quantum state, which is proposed to be the origin of spontaneous gauge symmetry breaking, for which nonconservation of particle number N is not essential. The approach based on nonvanishing expectation of the field operator, $$, is only a coherent-state approximation in thermodynamic limit. When $N \to \infty$, this approach is equivalent, but $$ is not the macroscopic wavefunction.
Adding a gauge symmetry breaking field $\ensuremath{-}\ensuremath{\nu}\sqrt{V}({a}_{0}+{a}_{0}^{*})$ to the Hamiltonian of some simplified models of an interacting Bose gas we compute the condensate density and the symmetry breaking order parameter in the limit of infinite volume and prove Bogoliubov's asymptotic hypothesis ${\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}⟨{a}_{0}⟩∕\sqrt{V}=\mathrm{sgn}\phantom{\rule{0.2em}{0ex}}\ensuremath{\nu}\phantom{\rule{0.2em}{0ex}}{\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}\sqrt{⟨{a}_{0}^{*}{a}_{0}⟩∕V}$ where the averages are taken in the ground state or in thermal equilibrium states. Letting $\ensuremath{\nu}$ tend to zero in this equation we obtain that Bose-Einstein condensation occurs if and only if the gauge symmetry is spontaneously broken. The simplification consists in dropping the off-diagonal terms in the momentum representation of the pair interaction. The models include the mean field and the imperfect (Huang-Yang-Luttinger) Bose gas. An implication of the result is that the compressibility sum rule cannot hold true in the ground state of the one-dimensional mean-field Bose gas. Our method is based on a resolution of the Hamiltonian into a family of single-mode (k=0)(\mathbf{k}=0)(k=0) Hamiltonians and on the analysis of the associated microcanonical ensembles.
Abstract In this chapter we outline the microscopic theory of an interacting Bose fluid. The aim is to provide a basis for a microscopic understanding of the excitations in superfluid and normal 4He discussed in the following chapters. The microscopic picture can be omitted on a first reading and referred to when desired.We assume that the reader is familiar with the theory of Bose-Einstein condensation in a Bose gas introduced in §7.2 and presented in many texts on statistical mechanics. In §7.2 we also discussed the generalization of Bose condensation to an interacting fluid by Penrose and Onsager (1956).Condensation was identified with macroscopic occupation of one of the eigenstates of the fluid and the appearance of an infinitely long tail in the one-body density matrix, denoted as off-diagonal long range order (ODLRO). For a homogeneous fluid it is natural to take momentum states as the eigenstates and to assume that the macroscopically occupied state is the state with momentum p = 0. In this case the criterion reduces to that for a gas but with a firm foundation.
This chapter addresses the fundamental role of order parameters and symmetry breaking in characterizing phases and understanding phase transitions and argues that the origin of non-analyticity is in fact the thermodynamic limit. It explores the Landau definition that a transition is first order if the order parameter is discontinuous at the transition point and second order if it is continuous. The chapter presents a general theory which in leading approximation embodies the physics of the Landau function of the order parameter. The mean field theory may be applied quite generally to any system for which the local order parameter has been identified. Once the order parameter for Bose condensation is recognized, the order parameter for superconducting or superfluid transitions follow naturally. For superconducting metals, pairs of electrons at the Fermi surface form bound states which behave effectively as Bosons and thus undergo Bose condensation.