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

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

2007-02-05

arxiv.org/abs/cond-mat/0702058v5

Physics

Condensed Matter

Statistical Mechanics

major revision of previous version; 4.2 pages, 3 figures

Scientific paper

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.

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