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

Abstract

As is well-known, in Bogoliubov's theory of an interacting Bose gas the ground state of the Hamiltonian $\hat{H}={\sum }_{\mathbf{k}\ne 0}{\hat{H}}_{\mathbf{k}}$ is found by diagonalizing each of the Hamiltonians ${\hat{H}}_{\mathbf{k}}$ corresponding to a given momentum mode $\mathbf{k}$ independently of the Hamiltonians ${\hat{H}}_{{\mathbf{k}}^{\prime }(\ne \mathbf{k})}$ 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 ${\hat{H}}_{\mathbf{k}}$ are diagonalized are not disjoint, but have the $\mathbf{k}=0$ 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.

arXiv commentmajor revision of previous version; 4.2 pages, 3 figures