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arXiv:math-ph/0412056 · Physical Review Letters

Equivalence of Bose-Einstein Condensation and Symmetry Breaking

András Sütö

Published Mar 2, 2005• 30 citations
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Abstract

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.

DOI
10.1103/physrevlett.94.080402

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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.
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 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.
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.
In this work we show that the introduction of a U(1) symmetry breaking field in the energy operator of the boson-free gas, is equivalent, in the thermodynamic limit, to the inclusion, in the Hamiltonian of the ideal gas, of a non-linear function of the number operator associated with the zero mode. In other words, the limit pressures coincide. Moreover, both models undergo non conventional Bose-Einstein condensation (BEC) for strictly negative values of the chemical potential μ.μ.μ. Finally, the proof of equivalence of limit pressures is extended to a class of full-diagonal models.