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arXiv:cond-mat/0703498 · Progress of Theoretical Physics

Re-Examining Bogoliubov's Theory of an Interacting Bose Gas

A. M. Ettouhami

Published Mar 1, 2012• 15 citations
arXivHTMLPDF

Abstract

As is well-known, in the conventional formulation of Bogoliubov's theory of an interacting Bose gas, the Hamiltonian is written as a decoupled sum of contributions from different momenta of the form . Then, each of the single-mode Hamiltonians is diagonalized separately, and the resulting ground state wavefunction of the total Hamiltonian is written as a simple product of the ground state wavefunctions of each of the single-mode Hamiltonians . While this way of diagonalizing the total Hamiltonian may seem to be valid from the perspective of the standard, number non-conserving Bogoliubov's method, where the k = 0 state is removed from the Hilbert space and hence the individual Hilbert spaces where the Hamiltonians {} are diagonalized are disjoint from one another, we argue that from a number-conserving perspective this diagonalization method may not be adequate since the true Hilbert spaces where the Hamiltonians {} should be diagonalized all have the k = 0 state in common, and hence the ground state wavefunction of the total Hamiltonian may not be written as a simple product of the ground state wavefunctions of the 's. In this paper, we give a thorough review of Bogoliubov's method, and discuss a variational and number-conserving formulation of this theory in which the k = 0 state is restored to the Hilbert space of the interacting gas, and where, instead of diagonalizing the Hamiltonians separately, we diagonalize the total Hamiltonian . When this is done, we find that the ground state energy is lowered below the Bogoliubov result, and the depletion of bosons is significantly reduced with respect to the one obtained in the number non-conserving treatment. We also find that the spectrum of the usual αk excitations of Bogoliubov's method changes from a gapless one, as predicted by the standard, number non-conserving formulation of this theory, to one which exhibits a finite gap in the k → 0 limit. We discuss the presence of a gap in the spectrum of the αk's in light of Goldstone's theorem, and show that there is no contradiction with the latter.

DOI
10.1143/ptp.127.453

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As is well-known, in the conventional formulation of Bogoliubov's theory of an interacting Bose gas, the Hamiltonian is written as a decoupled sum of contributions from different momenta of the form . Then, each of the single-mode Hamiltonians is diagonalized separately, and the resulting ground state wavefunction of the total Hamiltonian is written as a simple product of the ground state wavefunctions of each of the single-mode Hamiltonians . While this way of diagonalizing the total Hamiltonian may seem to be valid from the perspective of the standard, number non-conserving Bogoliubov's method, where the k = 0 state is removed from the Hilbert space and hence the individual Hilbert spaces where the Hamiltonians {} are diagonalized are disjoint from one another, we argue that from a number-conserving perspective this diagonalization method may not be adequate since the true Hilbert spaces where the Hamiltonians {} should be diagonalized all have the k = 0 state in common, and hence the ground state wavefunction of the total Hamiltonian may not be written as a simple product of the ground state wavefunctions of the 's. In this paper, we give a thorough review of Bogoliubov's method, and discuss a variational and number-conserving formulation of this theory in which the k = 0 state is restored to the Hilbert space of the interacting gas, and where, instead of diagonalizing the Hamiltonians separately, we diagonalize the total Hamiltonian . When this is done, we find that the ground state energy is lowered below the Bogoliubov result, and the depletion of bosons is significantly reduced with respect to the one obtained in the number non-conserving treatment. We also find that the spectrum of the usual αk excitations of Bogoliubov's method changes from a gapless one, as predicted by the standard, number non-conserving formulation of this theory, to one which exhibits a finite gap in the k → 0 limit. We discuss the presence of a gap in the spectrum of the αk's in light of Goldstone's theorem, and show that there is no contradiction with the latter.
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 re-examine the way in which Bogoliubov's theory of a dilute Bose gas at T=0T=0T=0 has been extended to describe the statistical mechanics of interacting bosons at finite temperature. We show explicitly that the field-theoretic calculation of the grand partition function in this formulation amounts to a canonical trace over the eigenfunctions of the Bogoliubov Hamiltonian at fixed total number of bosons NNN, and that the additional trace over NNN that is required in the grand-canonical formalism is never carried out. This implies that what usually passes as the grand-canonical treatment of the Bogoliubov Hamiltonian is not quite grand-canonical, and is in fact a canonical one. We also show that the discontinuity in the condensate density predicted by previous formulations of this theory as the temperature TTT goes past the critical transition temperature TcT_cTc​ is a direct consequence of an inappropriate generalization of the Bogoliubov prescription to finite temperatures, and that this discontinuity disappears when this prescription is either used as a zero temperature approximation or avoided altogether. Armed with the above findings, we reformulate the statistical mechanics of interacting bosons in the canonical ensemble and derive the thermodynamics of the system. We then show how the canonical treatment can be used to setup a truly grand-canonical description of the statistical mechanics of a weakly interacting Bose gas where the average number of bosons in the system varies with temperature, unlike existing formulations where the total number of bosons NNN is taken to be a constant that does not depend on TTT. Consequences on the physics of interacting bosons are briefly discussed.
We consider the Bogoliubov vacuum state in the number-conserving Bogoliubov theory proposed by Castin and Dum [Phys. Rev. A 57 , 3008 (1998)]. We show that, in the particle representation, the vacuum can be written in a simple diagonal form. The vacuum state can describe the stationary N -particle ground state of a condensate in a trap, but it can also represent a dynamical state when, for example, a Bose-Einstein condensate initially prepared in the stationary ground state is subject to a time-dependent perturbation. In both cases the diagonal form of the Bogoliubov vacuum can be obtained by basically diagonalizing the reduced single-particle density matrix of the vacuum. We compare N -body states obtained within the Bogoliubov theory with the exact ground states in a 3-site Bose-Hubbard model. In this example, the Bogoliubov theory fails to accurately describe the stationary ground state in the limit when N → ∞ but a small fraction of depleted particles is kept constant.
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.
The excitation spectrum for the three-dimensional Bose gas in the Bose-Einstein condensation phase is calculated nonperturbatively with the modified Hartree-Fock-Bogoliubov theory, which is both conserving and gapless. From improved $\ensuremath{\Phi}$-derivable theory, the diagrams needed to preserve the Ward-Takahashi identity are re-summed in a systematic and nonperturbative way. It is valid up to the critical temperature where the dispersion relation of the low-energy excitation spectrum changes from linear to quadratic. Because including the higher-order fluctuation, the results show significant improvement on the calculation of the shift of critical temperature with other conserving and gapless theories.
In this paper we consider an interacting Bose gas at zero temperature, in a finite box and in the mean field limiting regime. The N gas particles interact through a pair potential of positive type and with an ultraviolet cut-off. Its (nonzero) Fourier components are sufficiently large with respect to the corresponding kinetic energies of the modes. Using the multi-scale technique in the occupation numbers of particle states introduced in [Pi1], we provide a convergent expansion of the ground state of the particle number preserving Bogoliubov Hamiltonian in terms of the bare operators. In the limit N \to \infty the expansion is up to any desired precision.