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arXiv:quant-ph/9702026

Spontaneous Symmetry Breaking, Off-diagonal Long-range Order, and Nucleation of Quantum State

Yu Shi

Published Feb 11, 1997• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/9702026

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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.
In many statistical models (but not all) one can introduce a local order parameter associated with a finite, discrete or continuous group of symmetries. The higher the temperature, the more important the fluctuations. One expects therefore in general to find pure phases at low temperature, with a reduced symmetry group, as a consequence of a nonvanishing expectation value of some order parameter. This situation is referred to as spontaneous symmetry breaking. A typical example is the classical Heisenberg model describing short-range interactions of an n -vector field ϕ, with an orthogonal O(n) symmetry group. For n = 1, 2, 3, this can account for a Curie transition from a ferromagnetic to a paramagnetic phase. In particle physics the σ-model of Gell-Mann and Levy involves a spontaneous symmetry breaking of chiral invariance, typical of massless spinor fields, accompanied by soft excitation modes, the so-called Goldstone modes, associated to a π-meson triplet and leading to a nonvanishing dynamical fermion mass. In a first and rather crude approximation, one can analyse the action itself or an effective one incorporating some fluctuation effects, and look for extrema as a function of field configurations, generally translationally invariant. The remaining fluctuations are then treated perturbatively. This mean field method is common to a great variety of domains, ranging from the Clausius–Mossotti formula for a polarizable medium, the Weiss molecular field in the theory of magnetism, Landau's effective action in various statistical contexts, the effective medium approximation in disordered systems, the Hartree–Fock method in atomic or many-body physics, to the semiclassical approximation in the study of quantum systems.
Since the mechanism of spontaneous symmetry breaking has such a relevant impact for the description of physical systems, its explanation should be based on physical considerations with a direct operational meaning. To this purpose, it is useful to recognize that the operational/experimental description of a physical system (not necessarily quantum!) involves the following elements: the set A of its measurable quantities, briefly called observables ; the measurement of the time evolution of the observables; the set ∑ of configurations or states in which the system may be prepared, according to well defined protocols of experimental preparations; the measurement of an observable A , when the system is in the state ω is operationally defined by the experimental expectation 〈A〉 ω
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 broken symmetry state with off-diagonal long-range order (ODLRO), which is characterized by the vacuum expectation value of the operator of creation of the conserved quantum number Q, has the time-dependent order parameter. However, the breaking of the time translation symmetry is observable only if the charge Q is not strictly conserved and may decay. This dichotomy is resolved in systems with quasi-ODLRO. These systems have two well separated relaxation times: the relaxation time τ Q of the charge Q and the energy relaxation time τ E . If τ Q ≫ τ E , the perturbed system relaxes first to the state with the ODLRO, which persists for a long time and finally relaxes to the full equilibrium static state. In the limit τQ → ∞, but not in the strict limit case when the charge Q is conserved, the intermediate ODLRO state can be considered as the ground state of the system at fixed Q with the observable spontaneously broken time translation symmetry. Examples of systems with quasi-ODLRO are provided by superfluid phase of liquid 4He, Bose-Einstein condensation of magnons (phase coherent spin precession) and precessing vortices.
We consider the self-evolution of strongly nonequilibrium interacting Bose gas. Because of the mere fact of large (relative to unity) occupation numbers in the initial state the problem is directly reduced to the question of temporal evolution of the statistical matrix diagonal in the coherent-state representation. Strictly speaking, gauge symmetry is not destroyed even when the long-range ordering is completed, owing to the inevitable averaging over the ensemble. Actual symmetry breaking can occur only as a result of introducing a small term of the Hamiltonian violating conservation of particles, or quantum-mechanical measurement, also implying nonconservation of particles.