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Physical review. B, Condensed matter

Unified symmetry-breaking theory of Bose-Einstein condensation in superfluids

A. C. Olinto

Published Apr 1, 1991• 1 citation
Publisher

Abstract

The usual symmetry-breaking procedures for Bose condensed systems, namely, the Bogoliubov prescription, the symmetry-breaking term added to the Hamiltonian, and the canonical shift transformation are unified into a single formalism. By taking into account the condensate reservoir as a source and sink of excited particles, exact Ward identities are solved in the shielded-potential approximation. A relationship between the condensate density ${\mathit{n}}_{0}$ and the superfluid density ${\mathit{n}}_{\mathit{S}}$ is obtained in closed form. It is shown that the Bogoliubov prescription yields ${\mathit{n}}_{0}$\ensuremath{\simeq}${\mathit{n}}_{\mathit{S}}$ and ${\mathit{nU}}_{0}$\ensuremath{\ll}\ensuremath{\Vert}\ensuremath{\mu}\ensuremath{\Vert}, where n is the total density, ${\mathit{U}}_{0}$ the interaction constant, and \ensuremath{\mu} the chemical potential. On the other hand, for the canonical shift transformation one has ${\mathit{n}}_{0}$\ensuremath{\ll}${\mathit{n}}_{\mathit{S}}$ and ${\mathit{nU}}_{0}$\ensuremath{\gg}\ensuremath{\Vert}\ensuremath{\mu}\ensuremath{\Vert}. The latter, applied to superfluid $^{4}\mathrm{He}$ at saturated vapor pressure, gives excellent agreement between theory and experiment, without any adjustable parameter. The condensate density turns out to be strongly dependent on pressure as observed experimentally. The formalism provides in a natural way a consistent description of Bose systems in arbitrary D-dimensional space.

DOI
10.1103/physrevb.43.7737

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The usual symmetry-breaking procedures for Bose condensed systems, namely, the Bogoliubov prescription, the symmetry-breaking term added to the Hamiltonian, and the canonical shift transformation are unified into a single formalism. By taking into account the condensate reservoir as a source and sink of excited particles, exact Ward identities are solved in the shielded-potential approximation. A relationship between the condensate density n0{\mathit{n}}_{0}n0​ and the superfluid density nS{\mathit{n}}_{\mathit{S}}nS​ is obtained in closed form. It is shown that the Bogoliubov prescription yields n0{\mathit{n}}_{0}n0​\ensuremath{\simeq}nS{\mathit{n}}_{\mathit{S}}nS​ and nU0{\mathit{nU}}_{0}nU0​\ensuremath{\ll}\ensuremath{\Vert}\ensuremath{\mu}\ensuremath{\Vert}, where n is the total density, U0{\mathit{U}}_{0}U0​ the interaction constant, and \ensuremath{\mu} the chemical potential. On the other hand, for the canonical shift transformation one has n0{\mathit{n}}_{0}n0​\ensuremath{\ll}nS{\mathit{n}}_{\mathit{S}}nS​ and nU0{\mathit{nU}}_{0}nU0​\ensuremath{\gg}\ensuremath{\Vert}\ensuremath{\mu}\ensuremath{\Vert}. The latter, applied to superfluid 4He^{4}\mathrm{He}4He at saturated vapor pressure, gives excellent agreement between theory and experiment, without any adjustable parameter. The condensate density turns out to be strongly dependent on pressure as observed experimentally. The formalism provides in a natural way a consistent description of Bose systems in arbitrary D-dimensional space.
A gauge-symmetry-breaking parameter associated with the fluctuation of the zero-mode particles renders a source term to the continuity equation of interacting Bose systems. As a consequence, additional terms appear in the generalized Ward identities, which modify in an essential way the relation between the superfluid and condensate densities. An energy gap results as a function of the symmetry-breaking parameter and interatomic potential. The general results are worked out in the shielded potential approximation, and an equation for the condensate fraction n0{n}_{0}n0​(T) is obtained in closed form. At absolute zero the condensate fractions in dilute Bose systems and superfluid 4He^{4}\mathrm{He}4He are found to be n0{n}_{0}n0​=0.438 and n0{n}_{0}n0​=0.126, respectively.
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 idea of replacing the condensate operators by the ccc number N012{N}_{0}^{\frac{1}{2}}N021​​ in treating condensed Bose systems is known as Bogoliubov prescription (BP). Once BP is adopted, there are two ways to proceed. The first is the popular method of Hugenholtz and Pines, which we call $\ensuremath{\mu}$-fixed formalism. The second may be found in the well-known works of Foldy and Brueckner on a dense charged boson gas. Unlike the $\ensuremath{\mu}$-fixed formalism, the latter method does not require the introduction of the chemical potential, and thus it is called the NNN-fixed formalism. We argue that BP is rigorous in the $\ensuremath{\mu}$-fixed formalism but is not in the NNN-fixed one. BP in the NNN-fixed one gives correct results for terms of leading orders (Foldy and Brueckner took just these terms) but incorrect results for terms of higher orders.
Based on a classic paper by Ginibre [Commun. Math. Phys. 8, 26 (1968)] it is shown that, whenever Bogoliubov's approximation, that is, the replacement of a(0) and a*(0) by complex numbers in the Hamiltonian, asymptotically yields the right pressure, it also implies the asymptotic equality of |a(0)|(2)/V and a*(0)a(0)/V in symmetry breaking fields, irrespective of the existence or absence of Bose-Einstein condensation. Because the former was proved by Ginibre to hold for absolutely integrable superstable pair interactions, the latter is equally valid in this case. Apart from Ginibre's work, our proof uses only a simple convexity inequality due to Griffiths.