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Physical Review

Excitation Spectrum in Many-Boson Systems

Fumihiko Takano

Published Jul 15, 1961• 31 citations
Publisher

Abstract

When only two-body correlations are fully taken into account, there appears an energy gap in the excitation spectrum for many-boson systems as shown by Girardeau and Arnowitt and confirmed by Wentzel. This energy gap is shown to disappear and the spectrum to become phononlike again and proportional to the momentum for small momentum, if we construct the eigenmodes of excitations (collective excitations), taking into consideration appropriate higher-order terms.

DOI
10.1103/physrev.123.699

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When only two-body correlations are fully taken into account, there appears an energy gap in the excitation spectrum for many-boson systems as shown by Girardeau and Arnowitt and confirmed by Wentzel. This energy gap is shown to disappear and the spectrum to become phononlike again and proportional to the momentum for small momentum, if we construct the eigenmodes of excitations (collective excitations), taking into consideration appropriate higher-order terms.
The many-boson system with repulsive interactions is treated by a variational method based on a variational trial state of an exponential pair-excitation type obtained by a generalization of that of Bogoliubov; the treatment is closely related to an intermediate-coupling approximation with respect to pairs. The nonlinear integral equation which determines the variational ground state is derived, and various properties of this ground state and the associated excited states are examined. The resultant low-lying spectrum lies below that of Bogoliubov by an amount proportional to the total number of particles. The variational principle is shown to produce rigorous energy eigenvalue differences for the pair part of the Hamiltonian. The variational states, however, still exhibit unphysical features characteristic of pair-excitation states: The pair correlation function does not go to zero at zero particle separation and the phonon spectrum exhibits a gap above the ground state. It is suggested that these features can be removed by using states which take into account excitation of momentum-conserving groups of more than two particles.
The spectrum of phonon-like collective excitations in the system of Bose-atoms in optical lattice (more generally, in the system of quantum particles described by the Bose-Hubbard model) is investigated.Such excitations appear due to displacements of particles with respect to their local equilibrium positions.The two-level model taking into account the transitions of bosons between the ground state and the first excited state in potential wells, as well as interaction between them, is used.Calculations are performed within the random phase approximation in the hard-core boson limit.It is shown that excitation spectrum in normal phase consists of the one exciton-like band, while in the phase with BE condensate an additional band appears.The positions, spectral weights and widths of bands strongly depend on chemical potential of bosons and temperature.The conditions of stability of a system with respect to the lowering of symmetry and displacement modulation are discussed.