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arXiv:1108.3723 · Europhysics Letters (EPL)

Exact equivalence between one-dimensional Bose gases interacting via hard-sphere and zero-range potentials

Manuel Valiente

Published Apr 1, 2012• 23 citations
arXivHTMLPDF

Abstract

We prove the equivalence between the hard-sphere Bose gas and a system with momentum-dependent zero-range interactions in one spatial dimension, which we call extended hard-sphere Bose gas. The two-body interaction in the latter model has the advantage of being a regular pseudopotential. The most immediate consequence is the existence of its Fourier transform, permitting the formulation of the problem in momentum space, not possible with the original hard-core interaction. In addition, in the extended system, interactions are defined in terms of the scattering length, positive or negative, identified with the hard-sphere diameter only when it is positive. We are then able to obtain, directly in the thermodynamic limit, the ground-state energy of the strongly repulsive Lieb-Liniger gas and, more importantly, the energy of the lowest-lying super Tonks-Girardeau gas state with finite, strongly attractive interactions, in perturbation theory from the novel extended hard-sphere Bose gas.

DOI
10.1209/0295-5075/98/10010

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We prove the equivalence between the hard-sphere Bose gas and a system with momentum-dependent zero-range interactions in one spatial dimension, which we call extended hard-sphere Bose gas. The two-body interaction in the latter model has the advantage of being a regular pseudopotential. The most immediate consequence is the existence of its Fourier transform, permitting the formulation of the problem in momentum space, not possible with the original hard-core interaction. In addition, in the extended system, interactions are defined in terms of the scattering length, positive or negative, identified with the hard-sphere diameter only when it is positive. We are then able to obtain, directly in the thermodynamic limit, the ground-state energy of the strongly repulsive Lieb-Liniger gas and, more importantly, the energy of the lowest-lying super Tonks-Girardeau gas state with finite, strongly attractive interactions, in perturbation theory from the novel extended hard-sphere Bose gas.
Recent theoretical and experimental results demonstrate a close connection between the super-Tonks-Girardeau (STG) gas and a one-dimensional (1D) hard-sphere Bose (HSB) gas with hard-sphere diameter nearly equal to the 1D scattering length a1D{a}_{1\mathrm{D}}a1D​ of the STG gas, a highly excited gaslike state with nodes only at interparticle separations $|{x}_{j\ensuremath{\ell}}|={x}_{\mathrm{node}}\ensuremath{\approx}{a}_{1\mathrm{D}}$. It is shown herein that when the coupling constant gB{g}_{B}gB​ in the Lieb-Liniger interaction ${g}_{B}\ensuremath{\delta}({x}_{j\ensuremath{\ell}})$ is negative and $|{x}_{12}|\ensuremath{\geqslant}{x}_{\mathrm{node}}$, the STG and HSB wave functions for N=2N=2N=2 particles are not merely similar, but identical; the only difference between the STG and HSB wave functions is that the STG wave function allows a small penetration into the region ∣x12∣<xnode|{x}_{12}|<{x}_{\mathrm{node}}∣x12​∣<xnode​, whereas for a HSB gas with hard-sphere diameter ahs=xnode{a}_{\mathrm{hs}}={x}_{\mathrm{node}}ahs​=xnode​, the HSB wave function vanishes when ∣x12∣<ahs|{x}_{12}|<{a}_{\mathrm{hs}}∣x12​∣<ahs​. Arguments are given suggesting that the same theorem holds also for N>2N>2N>2. The STG and HSB wave functions for N=2N=2N=2 are given exactly in terms of a parabolic cylinder function, and for $N\ensuremath{\geqslant}2$, xnode{x}_{\mathrm{node}}xnode​ is given accurately by a simple parabola. The metastability of the STG phase generated by a sudden change of the coupling constant from large positive to large negative values is explained in terms of the very small overlap between the ground state of the Tonks-Girardeau gas and collapsed cluster states.
One-dimensional Bose gases are considered, interacting either through the hard-core potentials or through the contact delta potentials. Interest in these gases gained momentum because of the recent experimental realization of quasi-one-dimensional Bose gases in traps with tightly confined radial motion, achieving the Tonks-Girardeau (TG) regime of strongly interacting atoms. For such gases the Fermi-Bose mapping of wavefunctions is applicable. The aim of the present communication is to give a brief survey of the problem and to demonstrate the generality of this mapping by emphasizing that: (i) It is valid for nonequilibrium wavefunctions, described by the time-dependent Schrödinger equation, not merely for stationary wavefunctions. (ii) It gives the whole spectrum of all excited states, not merely the ground state. (iii) It applies to the Lieb-Liniger gas with the contact interaction, not merely to the TG gas of impenetrable bosons.
The system under consideration is an NNN-particle quantum-mechanical system enclosed in a volume VVV, in which the particles interact via two-body hard-sphere potentials, with hard-sphere diameter aaa. The two-body hard-sphere problem is first discussed by a generalization of Fermi's pseudopotential by means of which the problem is formulated entirely in terms of the scattering phase shifts. It is then shown that a pseudopotential for the NNN-body problem can be introduced, and leads to an expansion of the energy levels of the system in powers of aaa. The first order energy levels of a Bose and a Fermi system are calculated. For the Bose system, the first order energy formula exhibits an "energy gap" above the ground state, leading to properties of the system not dissimilar to that of a superfluid. The ground state energy for a Bose system is calculated to order a3{a}^{3}a3 and that for the Fermi system, to order a2{a}^{2}a2. The physical interpretation and validity of these results are discussed.
An NNN-body Bose-Einstein system of particles with long-range attraction and hard-sphere repulsion between particles is considered. It is shown that if the constants of the interaction have values within a certain range it is possible to calculate the ground-state energy of the system as a function of $\frac{\ensuremath{\Omega}}{N}$, where $\ensuremath{\Omega}$ is the volume of the box containing the system, in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\Omega}\ensuremath{\rightarrow}\ensuremath{\infty}$, with $\frac{N}{\ensuremath{\Omega}}=\ensuremath{\rho}$ fixed. The results show that the system can possess an NNN-body bound state, which has an equilibrium density and negative energy, and that the interactions can be saturating. Excited states are also considered. It is shown that low-lying excitations consist purely of phonons, whose velocity agrees with that computed from the macroscopic compressibility, furnished by the ground-state energy. The formula for the general excited energy levels suggests that thermodynamically the system may have a "gas" phase and two "liquid" phases, the transition between the two "liquid" phases being the analog of the Bose-Einstein condensation of the ideal gas. Thermodynamic considerations are, however, not contained in this paper.
Abstract The coordinate Bethe ansatz can be extended to a model, the Lieb–Liniger model, of a one-dimensional gas of Bosons interacting with repulsive δ-function potentials. It has attracted attention due to its relevance for experimental developments in the fields of ultracold gases and optical lattices. This chapter provides an exposition of the related classical nonlinear Schrödinger equation, followed by its generalization to the quantum model. It explores a limiting case, the Tonks-Girardeau gas. The δ-function potentials supply a kind of boundary condition on the wave functions allowing us to analyze the eigenfunctions of the Bethe ansatz, which are examined on the infinite line and for periodic boundary conditions. The latter leads to the Bethe ansatz equations. The solution of these equations is achieved in the thermodynamic limit for the ground state and for low-lying excited states.
A many-body pseudopotential for a hard-sphere system which is mathematically equivalent to the boundary conditions imposed on the wave functions is obtained as a generalization of Huang and Yang's binary pseudopotential, and the lowest order term of the ground state energy resulting from the triple collision is calculated for the Bose system. Instead of the direct perturbational treatment of the pseudopotential, the use of the ρk-representation for the Bose system is proposed and the ground state energy is calculated in this representation. It is shown that the energy thus calculated is of the form of an expansion in powers of (ρa3)1/2, where ρ is the average density and a the hard-sphere diameter, which is agreement with Lee and Yang's conclusion drawn from the binary collision expansion method. Some discussions associated with the roton spectrum of liquid He4 are given.