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arXiv:1204.4081 · Physical Review A

Equation of state for the one-dimensional attractive δ -potential Bose gas in the weak-coupling regime

Tsubasa Ichikawa, Izumi Tsutsui, Nobuhiro Yonezawa

Published Jul 19, 2012• 4 citations
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Abstract

Approximated formulas for real quasimomentum and the associated energy spectrum are presented for one-dimensional Bose gas with weak attractive contact interactions. On the basis of the energy spectrum, we obtain the equation of state in the high-temperature region, which is found to be the van der Waals equation without volume correction.

DOI
10.1103/physreva.86.015602

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Approximated formulas for real quasimomentum and the associated energy spectrum are presented for one-dimensional Bose gas with weak attractive contact interactions. On the basis of the energy spectrum, we obtain the equation of state in the high-temperature region, which is found to be the van der Waals equation without volume correction.
We study how the introduction of a two-body interaction changes the thermodynamic properties of the free Bose gas in a weak (scaled) external field. We show that, in the van der Waals limit, the thermodynamic pressure always is a ${C}^{\ensuremath{\infty}}$ function of the density in one and two dimensions, while this is not so in higher dimensions. The phase transition to the superfluid phase in the latter case is discussed in more detail by looking at the local densities. We also show that the van der Waals limit for this system does not correspond to the usual mean-field approximation.
We present a simple approximation scheme for the solution of the integral equations resulting from the Bethe-ansatz diagonalization of the one-dimensional Fermi gas with \ensuremath{\delta}-function attraction. These equations arise for the Hubbard model with attractive interaction in the limit of weak coupling and low density. We obtain the ground-state energy as a function of coupling and density in very good agreement with numerical solutions, as well as a value for the parameter determining the gap in the magnetic excitation spectrum which strongly supports a conjecture of Larkin and Sak.
Within the self-consistent Hartree–Fock approximation, the equilibrium weakly nonideal Bose gas with a delta-shaped interaction potential in the presence of the Bose–Einstein condensate is considered without using quasi-averages. On this basis, using the virial theorem and diagram techniques of the perturbation theory for the equilibrium system in a macroscopic volume, the equation of state providing the isothermal compressibility finiteness, including the Bose–Einstein condensate domain of existence, is obtained.
In this paper, we have studied the one-dimensional Fermi gas with attractive δ interaction potential in the quasi-particle random phase approximation at zero temperature. Using the theoretical formalism of Alm and Schuck, we have studied the behavior of collective modes over the whole coupling regime from weak coupling (high density) to strong coupling (low density). We observe that in the weak- coupling limit particle-hole RPA is approached to low momenta and the collective mode in the strong coupling limit reproduces the Bogoliubov mode for the weakly interacting bosons.