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Cambridge University Press eBooks

Broken Gauge Symmetry in a Bose Condensate

A. J. Leggett

Published Apr 6, 1995• 23 citations
Publisher

Abstract

This paper examines the meaning of the “phase” of a Bose condensate, and in particular the degree of validity of the often used analogy with the direction of magnetization of a ferromagnet. It focusses on two specific questions: (i) Under what circumstances is the relative phase of two condensates well defined? (ii) Would it be possible in principle to set up a “standard of phase”? In the most obvious sense, the answer to (ii) is concluded to be no. As is well known, it is very fashionable nowadays to treat Bose condensation as a special case of the more general idea of spontaneously broken symmetry, which is ubiquitous in condensed-matter physics [1]. The standard account goes something like this: Just as in a magnetic material described by an isotropic Heisenberg model, the Hamiltonian is invariant under simultaneous rotation of all the spins, so in a Bose system described by the standard creation and annihilation operators ψ † (r), ψ(r) it is invariant under the global U (1) gauge transformation ψ(r) → ψ(r)e iϕ , ψ † (r) → ψ † (r)e −iϕ . Thus, at first sight, symmetry forbids either the expectation value < S > of the magnetization of the magnetic material, or the corresponding quantity < ψ > in the Bose system, to take a finite value.

DOI
10.1017/cbo9780511524240.021

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Physical Review A•Jun 1, 1996•46 citations
This paper examines the meaning of the “phase” of a Bose condensate, and in particular the degree of validity of the often used analogy with the direction of magnetization of a ferromagnet. It focusses on two specific questions: (i) Under what circumstances is the relative phase of two condensates well defined? (ii) Would it be possible in principle to set up a “standard of phase”? In the most obvious sense, the answer to (ii) is concluded to be no. As is well known, it is very fashionable nowadays to treat Bose condensation as a special case of the more general idea of spontaneously broken symmetry, which is ubiquitous in condensed-matter physics [1]. The standard account goes something like this: Just as in a magnetic material described by an isotropic Heisenberg model, the Hamiltonian is invariant under simultaneous rotation of all the spins, so in a Bose system described by the standard creation and annihilation operators ψ † (r), ψ(r) it is invariant under the global U (1) gauge transformation ψ(r) → ψ(r)e iϕ , ψ † (r) → ψ † (r)e −iϕ . Thus, at first sight, symmetry forbids either the expectation value < S > of the magnetization of the magnetic material, or the corresponding quantity < ψ > in the Bose system, to take a finite value.
The question of whether Bose-Einstein condensation involves spontaneous symmetry breaking is surprisingly controversial. We review the theory of spontaneous symmetry breaking in ferromagnets, compare it with the theory of symmetry breaking in condensates, and discuss the different viewpoints on the correspondence to experiments. These viewpoints include alternative perspectives in which we can treat condensates with fixed particle numbers, and where coherence arises from measurements. This question relates to whether condensates of quasiparticles such as polaritons can be viewed as “real” condensates. Introduction Spontaneous symmetry breaking is a deep subject in physics with long historical roots. At the most basic level, it arises in the field of cosmology. Physicists have long had an aesthetic principle that leads us to expect symmetry in the all of the basic equations of physical law. Yet the universe is manifestly full of asymmetries. How does a symmetric system acquire asymmetry merely by evolving in time? Starting in the 1950s, cosmologists began to borrow the ideas of spontaneous symmetry breaking from condensed matter physics, which were originally developed to explain spontaneous magnetization in ferromagnetic systems. Spontaneous coherence in all its forms (e.g., Bose-Einstein condensation [BEC], superconductivity, and lasing) can be viewed as another type of symmetry breaking. The Hamiltonian of the system is symmetric, yet under some conditions, the energy of the system can be reduced by putting the system into a state with asymmetry, namely, a state with a common phase for a macroscopic number of particles. The symmetry of the system implies that it does not matter what the exact choice of that phase is, as long as it is the same for all the particles. It is not obvious, however, whether the symmetry breaking which occurs in spontaneous coherence of the type seen in lasers or in Bose-Einstein condensation is the same as that seen in ferromagnetic systems. There are similarities in the systems which encourage the same view of all types of symmetry breaking, but there are also differences. In fact, there is a substantial school of thought that symmetry breaking in Bose-Einstein condensates with ultracold atoms is a “convenient fiction” (a term applied to optical coherence by Mølmer [1]).
The interplay between spontaneously broken gauge symmetries and Bose–Einstein condensation has long been controversially discussed in science, since the equations of motion are invariant under phase transformations. Within the present model, it is illustrated that spontaneous symmetry breaking appears as a non-local process in position space, but within disjoint subspaces of the underlying Hilbert space. Numerical simulations show that it is the symmetry of the relative phase distribution between condensate and non-condensate quantum fields which is spontaneously broken when passing the critical temperature for Bose–Einstein condensation. Since the total number of gas particles remains constant over time, the global U(1)-gauge symmetry of the system is preserved.
The interplay between spontaneously broken gauge symmetries and Bose-Einstein condensation has long been controversially discussed in science, since the equation of motions are invariant under phase transformations. Within the present model it is illustrated that spontaneous symmetry breaking appears as a non-local process in position space, but within disjoint subspaces of the underlying Hilbert space. Numerical simulations show that it is the symmetry of the relative phase distribution between condensate and non-condensate quantum fields which is spontaneously broken when passing the critical temperature for Bose-Einstein condensation. Since the total number of gas particles remains constant over time, the global U(1)-gauge symmetry of the system is preserved.
A renormalizable parameter \ensuremath{\xi} (0\ensuremath{\le}\ensuremath{\xi}\ensuremath{\le}1) is introduced into the interacting Bose-system Hamiltonian such that the zero-mode amplitudes satisfy the commutation relation [b0{b}_{0}b0​,${b}_{0}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$]=1-\ensuremath{\xi}. The different phases of the system correspond to specific values of \ensuremath{\xi}. The normal component (phase I) is obtained when \ensuremath{\xi}=0 and there are two ordered states according to the strength of the interaction: Phase II' (0<\ensuremath{\xi}\ensuremath{\ll}1) is found in the strongly interacting system and is characterized by a broken gauge symmetry without Bose-Einstein condensation, while phase II'' (\ensuremath{\xi}=1), in the weakly interacting system, displays Bose-Einstein condensation. The continuity equation for the particle density holds in both ordered phases to within an infinitesimal term. A gauge transformation on the field operators shows that \ensuremath{\xi} ''twists'' the phase angle related to the zero mode, and a response to this twisting field is defined in terms of the generalized force \ensuremath{\chi}(T) associated with \ensuremath{\xi}. For the ideal Bose gas \ensuremath{\chi}(T) equals the condensate density and hence is simply related to the helicity modulus introduced by Fisher et al. A renormalization-group transformation shows that the dense system transition (I-II') gives rise to a two-thirds correlation-length critical exponent, whereas a unity crossover exponent appears in the dilute system transition (I-II''), in agreement with the experimental findings.
In this paper, we have used a technique , when a weak link is created between two systems which have undergone some type of gauge symmetry breaking. In fact the Josephson effect has been observed in superfluid .Bose condensate atomic gases is generally expected. A system whose gauge symmetry has been spontaneously broken can be described by order parameter that behaves in much respect like a macroscopic function)(rΨ. For Bose Einstein Condensates of dilute alkali gases )(rΨis the wave function of the macroscopically occupied one atomic state