The Gaussian Approximation to Homogeneous Bose Gas

Abstract

We study low-lying excitations of a spinless homogeneous Bose gas with repulsive interaction at zero temperature in terms of the Gaussian mean field approximation. The dynamical equations of this approximation have been derived for small displacements from the static Hartree-Fock-Bogoliubov solution. We obtain a gapped continuous band of excitations above a discrete branch with phonon behavior at long wavelength regime. We also discuss the available forms of excitations and conclude that there are constraints on the first order deviations of the Gaussian approximation parameters and they are generated by an infinitesimal unitary transformation.

Appendix: The Definitions of the Dynamics Matrix

The definitions of the matrices necessary to the dynamics equations (given in (20) and (21)) of the generalized averages are given below. The matrix A(k,q) is given by

(62)

and the B(k,q) is

(63)

where c(k)=cosh(2σ _k ) and s(k)=sinh(2σ _k ). The matrix I has just the element I ₁₁ different from zero and equal to 1. The matrix that couple the excitations to the condensate, C(k,q), is given by

$$ C(\mathbf{ k}, \mathbf{q})=\frac{\rho_{0}}{2} \biggl(v \biggl(\biggl\lvert \mathbf{k}+\frac{\mathbf{q}}{2} \biggr\rvert \biggr) C^+(\mathbf{ k},\mathbf{ q})+v \biggl(\biggl \vert \mathbf{k}-\frac{\mathbf{q}}{2} \biggr \vert \biggr) C^{-}(\mathbf{k},\mathbf{ q})+2 v(q) C^3(\mathbf{ k}, \mathbf{q}) \biggr) $$

(64)

where

(65)

(66)

and

(67)

The notation used to C ^±(k,q) was choice just to make the superscript sign of C ^±(k,q) coincides with the sign of $v (\lvert \mathbf{k}\pm\frac{\mathbf{q}}{2}\rvert )$.

Finally the matrices G(q) and L(k,q) are given by

$$ G({q})=\left(\begin{array}{c@{\quad}c} 0 & \varDelta({q})-\varLambda({q})\\[3pt] \varDelta({q})+\varLambda({q}) & 0 \end{array}\right), $$

(68)

and

(69)