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arXiv:cond-mat/0702058

Nature of the Bogoliubov ground state of a weakly interacting Bose gas

A. M. Ettouhami

Published Feb 5, 2007• 0 citations
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Abstract

As is well-known, in Bogoliubov's theory of an interacting Bose gas the ground state of the Hamiltonian $\hat{H}=\sum_{\bf k\neq 0}\hat{H}_{\bf k}$ is found by diagonalizing each of the Hamiltonians $\hat{H}_{\bf k}$ corresponding to a given momentum mode ${\bf k}$ independently of the Hamiltonians $\hat{H}_{\bf k'(\neq k)}$ of the remaining modes. We argue that this way of diagonalizing $\hat{H}$ may not be adequate, since the Hilbert spaces where the single-mode Hamiltonians $\hat{H}_{\bf k}$ are diagonalized are not disjoint, but have the ${\bf k}=0$ in common. A number-conserving generalization of Bogoliubov's method is presented where the total Hamiltonian $\hat{H}$ 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 $k\to 0$ limit.

DOI
10.48550/arxiv.cond-mat/0702058

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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.
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.
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.
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.
In this paper we introduce a novel multi-scale technique to study many-body quantum systems where the total number of particles is kept fixed. The method is based on Feshbach map and the scales are represented by occupation numbers of particle states. Here, we consider a three-modes (including the zero mode) Bogoliubov Hamiltonian for a sufficiently small ratio between the kinetic energy and the Fourier component of the (positive type) potential corresponding to the two nonzero modes. For any space dimension d\geq 1 and in the mean field limiting regime (i.e., at fixed box volume |Λ| and for a number of particles, N, sufficiently large) this method provides the construction of the ground state and its expansion in terms of the bare operators that in the limit N \to \infty is up to any desired precision. In space dimension d \geq 3 the method provides similar results for an arbitrarily large (finite) box and a large but fixed particle density ρ, i.e.,ρis independent of the size of the box.
The present short note is simply intended to communicate that I have analytically diagonalized the Bogoliubov truncated Hamiltonian HcH_cHc​~\cite{Bogo1,Bogo2}, in an interacting bosonic gas. This is the natural prosecution of my work~\cite{MS}, now denoted as (I), where the diagonalization was performed only in the subspace corresponding to zero momentum collective excitations (CE).