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arXiv:cond-mat/0608602

The Bose Gas Low Momentum Limit Revisited

Hugo Pérez Rojas, D. Oliva

Published Aug 28, 2006• 0 citations
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Abstract

We discuss the standard approach to the problem of the low momentum limit of the spectrum for a weakly interacting Bose gas. The Bogoliubov's spectrum is shown to be obtained as a Goldstone mode thanks to the introduction of a chemical potential $μ$. This procedure has, however, difficulties since the breaking of the gauge symmetry implies that the corresponding chemical potential must be taken as zero, unless it is introduced before breaking the symmetry. But if this is done, after the symmetry breaking $μ$ loses its meaning as a chemical potential. An alternative two-mode solution is suggested having two modes, one of them being the free-particle quadratic in momentum spectrum, the second bearing a gap. This gap leads to a $λ$-type behavior of the specific heat near the critical temperature.

DOI
10.48550/arxiv.cond-mat/0608602

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We discuss the standard approach to the problem of the low momentum limit of the spectrum for a weakly interacting Bose gas. The Bogoliubov's spectrum is shown to be obtained as a Goldstone mode thanks to the introduction of a chemical potential μμμ. This procedure has, however, difficulties since the breaking of the gauge symmetry implies that the corresponding chemical potential must be taken as zero, unless it is introduced before breaking the symmetry. But if this is done, after the symmetry breaking μμμ loses its meaning as a chemical potential. An alternative two-mode solution is suggested having two modes, one of them being the free-particle quadratic in momentum spectrum, the second bearing a gap. This gap leads to a λλλ-type behavior of the specific heat near the critical temperature.
Abstract We derive and analyze the coupled equations of quadratic approximation of the Bogoliubov model for a weakly interacting Bose gas. The first equation determines the condensate density as a variational parameter and ensures the minimum of the grand thermodynamic potential. The second one provides a relation between the total number of particles and chemical potential. Their consistent theoretical analysis is performed for a number of model interaction potentials including contact (local) and nonlocal interactions, where the latter provide nontrivial dependencies in momentum space. We demonstrate that the derived equations have no solutions for the local potential, although they formally reproduce the well-known results of the Bogoliubov approach. At the same time, it is shown that these equations have the solutions for physically relevant nonlocal potentials. We show that in the regimes close to experimental realizations with ultracold atoms, the contribution of the terms originating from the quadratic part of the truncated Hamiltonian to the chemical potential can be of the same order of magnitude as from its c -number part. Due to this fact, in particular, the spectrum of single-particle excitations in the quadratic approximation acquires a gap. The issue of the gap is also discussed.
A mechanism generating a λλλ-type behavior in the specific heat of a Bose gas near the critical temperature TcT_cTc​ is discussed. It is shown to work for a general class of quasiparticle spectra, and especially, for the Bogoliubov's model of a weakly interacting Bose gas, where a temperature-dependent quasi-particle spectrum is obtained.
Abstract: The one-loop effective potential for non-relativistic bosons with a delta function repulsive potential is calculated for a given chemical potential using functional methods. After renormalization and at zero temperature it reproduces the standard ground state energy and pressure as function of the particle density. At finite temperatures it is found necessary to include ring corrections to the one-loop result in order to satisfy the Goldstone theorem. It is natural to introduce an effective chemical potential directly related to the order parameter and which uniformly decreases with increasing temperatures. This is in contrast to the the ordinary chemical potential which peaks at the critical temperature. The resulting thermodynamics in the condensed phase at very low temperatures is found to be the same as in the Bogoliubov approximation where the degrees of freedom are given by the Goldstone bosons. At higher temperatures the ring corrections dominate and result in a critical temperature unaffected by the interaction. 1
As is well-known, in Bogoliubov's theory of an interacting Bose gas the ground state of the Hamiltonian H^=∑k≠0H^k\hat{H}=\sum_{\bf k\neq 0}\hat{H}_{\bf k}H^=∑k=0​H^k​ is found by diagonalizing each of the Hamiltonians H^k\hat{H}_{\bf k}H^k​ corresponding to a given momentum mode k{\bf k}k independently of the Hamiltonians H^k′(≠k)\hat{H}_{\bf k'(\neq k)}H^k′(=k)​ of the remaining modes. We argue that this way of diagonalizing H^\hat{H}H^ may not be adequate, since the Hilbert spaces where the single-mode Hamiltonians H^k\hat{H}_{\bf k}H^k​ are diagonalized are not disjoint, but have the k=0{\bf k}=0k=0 in common. A number-conserving generalization of Bogoliubov's method is presented where the total Hamiltonian H^\hat{H}H^ is diagonalized directly. When this is done, the spectrum of excitations changes from a gapless one, as predicted by Bogoliubov's method, to one which has a finite gap in the k→0k\to 0k→0 limit.
We discuss the mechanism of spontaneous symmetry breaking and the elementary excitations for a weakly-interacting Bose gas at a finite temperature. We consider both the non-relativistic case, described by the Gross-Pitaevskii equation, and the relativistic one, described by the cubic nonlinear Klein-Gordon equation. We analyze similarities and differences in the two equations and, in particular, in the phase and amplitude modes (i.e., Goldstone and Higgs modes) of the bosonic matter field. We show that the coupling between phase and amplitude modes gives rise to a single gapless Bogoliubov spectrum in the non-relativistic case. Instead, in the relativistic case the spectrum has two branches: one is gapless and the other is gapped. In the non-relativistic limit we find that the relativistic spectrum reduces to the Bogoliubov one. Finally, as an application of the above analysis, we consider the Bose-Hubbard model close to the superfluid-Mott quantum phase transition and we investigate the elementary excitations of its effective action, which contains both non-relativistic and relativistic terms.