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Chinese Physics

Spontaneous U(1) symmetry breaking and Bose–Einstein condensation

Qian Feng, HUANG HONG-BIN, Qi Guan-Xiao, Shen Cai-Kang

Published Jul 1, 2006• 6 citations
Publisher

Abstract

Based on Bogoliubov's truncated Hamiltonian H B for a weakly interacting Bose system, and adding a U(1) symmetry breaking term V 1/2 (λ a 0 +λ * a 0 + ) to H B , we show by using the coherent state theory and the mean-field approximation rather than the c-number approximations, that the Bose–Einstein condensation(BEC) occurs if and only if the U(1) symmetry of the system is spontaneously broken. The real ground state energy and the justification of the Bogoliubov c-number substitution are given by solving the Schrödinger eigenvalue equation and using the self-consistent condition.

DOI
10.1088/1009-1963/15/7/033

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Based on Bogoliubov's truncated Hamiltonian H B for a weakly interacting Bose system, and adding a U(1) symmetry breaking term V 1/2 (λ a 0 +λ * a 0 + ) to H B , we show by using the coherent state theory and the mean-field approximation rather than the c-number approximations, that the Bose–Einstein condensation(BEC) occurs if and only if the U(1) symmetry of the system is spontaneously broken. The real ground state energy and the justification of the Bogoliubov c-number substitution are given by solving the Schrödinger eigenvalue equation and using the self-consistent condition.
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.
The Bogoliubov Model of the Weakly Nonideal Bose Gas In this chapter we review the basic theory of Bose-Einstein condensation (BEC) of excitons. To start, this means we must review the basic theory of BEC of any kind of particle, the theory known as the Bogoliubov model, after the foundational contributions made by N. N. Bogoliubov. This model introduces all the strange things associated with BEC: spontaneous symmetry breaking, off-diagonal long-range order (ODLRO), macroscopic occupation of a single quantum state, etc. It is often said that physicists who spend years studying quantum mechanics eventually warp their intuition so much that things like ODLRO seem normal. It is well worth stepping back every now and then to think about just how strange a Bose condensate is. First, consider the idea of spontaneous symmetry breaking. Many systems exist in which the underlying physics, expressed in the Hamiltonian, do not favor one state over another. The a priori probability of occupation of the different states by a particle is equal, i.e., symmetric. Nevertheless, in some cases, thermodynamics requires that a macroscopic number of particles must somehow “choose” one of the states preferentially. If they did not, the system would not be in equilibrium, i.e, could not have a definable temperature. If the particles preferentially choose one state, then the underlying symmetry has been broken. In the context of magnets, the breaking of the symmetry of the magnet (to have all its constituent dipoles lined up one way and not another) is energetically favored, since there are terms in the Hamiltonian that give lower energy for aligned dipoles, e.g., μS 1 · S 2 .
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.