Re-examining Bogoliubov's theory of an interacting Bose gas

A. M. Ettouhami

Published 2007-03-19, updated 2012-08-22Version 3

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 $\hat{H} = \sum_{k\neq 0}\hat{H}_{k}$. Then, each of the single-mode Hamiltonians $\hat{H}_{k}$ is diagonalized separately, and the resulting ground state wavefunction of the total Hamiltonian $\hat{H}$ is written as a simple product of the ground state wavefunctions of each of the single-mode Hamiltonians $\hat{H}_{k}$. We argue that, from a number-conserving perspective, this diagonalization method may not be adequate since the true Hilbert spaces where the Hamiltonians ${\hat{H}_{k}}$ should be diagonalized all have the ${k}=0$ state in common, and hence the ground state wavefunction of the total Hamiltonian $\hat{H}$ may {not} be written as a simple product of the ground state wavefunctions of the $\hat{H}_{k}$'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 ${k}=0$ state is restored to the Hilbert space of the interacting gas, and where, instead of diagonalizing the Hamiltonians $\hat{H}_{k}$ separately, we diagonalize the total Hamiltonian $\hat{H}$ as a whole. 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 $\alpha_{k}$ 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 $k\to 0$ limit.