Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2007-03-19
Physics
Condensed Matter
Statistical Mechanics
Major revision of version 1; 41 pages, 16 figures
Scientific paper
As is well-known, in the conventional formulation of Bogoliubov's theory of an interacting Bose gas, the Hamiltonian $\hat{H}$ is written as a decoupled sum of contributions from different momenta of the form $\hat{H} = \sum_{\bf k\neq 0}\hat{H}_{\bf k}$. Then, each of the single-mode Hamiltonians $\hat{H}_{\bf k}$ is diagonalized separately, and the resulting ground state wavefunction is written as a simple product of the ground state wavefunctions of each of the single-mode Hamiltonians $\hat{H}_{\bf k}$. We argue that, while this way of diagonalizing the total Hamiltonian $\hat{H}$ may seem to be valid from the perspective of the standard, number non-conserving Bogoliubov's method, where the $\bf k=0$ state is removed from the Hilbert space and hence the individual Hilbert spaces where the Hamiltonians $\{\hat{H}_{\bf k}\}$ are diagonalized are disjoint with one another, from a number-conserving perspective this diagonalization method may not be adequate since the true Hilbert spaces where the Hamiltonians $\{\hat{H}_{\bf k}\}$ should be diagonalized all have the ${\bf k}=0$ state in common, and hence the ground state wavefunction of the Hamiltonian $\hat{H}$ may {\em not} be written as a simple product of the ground state wavefunctions of the $\hat{H}_{\bf k}$'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 ${\bf k}=0$ state is restored to the Hilbert space of the interacting gas, and where, instead of diagonalizing the Hamiltonians $\hat{H}_{\bf k}$ separately, we diagonalize the total Hamiltonian $\hat{H}$ as a whole. When this is done, the spectrum of excitations of the system changes from a gapless one, as predicted by the standard, number non-conserving Bogoliubov theory, to one which exhibits a finite gap in the $k\to 0$ limit.
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