19 - Broken Gauge Symmetry in a Bose Condensate

Summary

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.