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arXiv:hep-th/9811247 · Physical Review Letters

Exact Solution of Double δ Function Bose Gas through an Interacting Anyon Gas

Anjan Kundu

Published Aug 16, 1999• 160 citations
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Abstract

A 1D Bose gas interacting through $\ensuremath{\delta},{\ensuremath{\delta}}^{\ensuremath{'}}$ and double $\ensuremath{\delta}$ function potentials is shown to be equivalent to a $\ensuremath{\delta}$ anyon gas, allowing an exact Bethe ansatz solution. In the noninteracting limit, it describes an ideal gas with generalized exclusion statistics and solves some recent controversies.

DOI
10.1103/physrevlett.83.1275

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A 1D Bose gas interacting through \ensuremathδ,\ensuremathδ\ensuremath′\ensuremath{\delta},{\ensuremath{\delta}}^{\ensuremath{'}}\ensuremathδ,\ensuremathδ\ensuremath′ and double \ensuremathδ\ensuremath{\delta}\ensuremathδ function potentials is shown to be equivalent to a \ensuremathδ\ensuremath{\delta}\ensuremathδ anyon gas, allowing an exact Bethe ansatz solution. In the noninteracting limit, it describes an ideal gas with generalized exclusion statistics and solves some recent controversies.
The low-energy properties of the one-dimensional anyon gas with a delta-function interaction are discussed in the context of its Bethe ansatz solution. It is found that the anyonic statistical parameter and the dynamical coupling constant induce Haldane exclusion statistics interpolating between bosons and fermions. Moreover, the anyonic parameter may trigger statistics beyond Fermi statistics for which the exclusion parameter alpha is greater than one. The Tonks-Girardeau and the weak coupling limits are discussed in detail. The results support the universal role of alpha in the dispersion relations.
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 gas of one-dimensional Bose particles interacting via a repulsive delta-function potential has been solved exactly. All the eigenfunctions can be found explicitly and the energies are given by the solutions of a transcendental equation. The problem has one nontrivial coupling constant, $\ensuremath{\gamma}$. When $\ensuremath{\gamma}$ is small, Bogoliubov's perturbation theory is seen to be valid. In this paper, we explicitly calculate the ground-state energy as a function of $\ensuremath{\gamma}$ and show that it is analytic for all $\ensuremath{\gamma}$, except $\ensuremath{\gamma}=0$. In Part II, we discuss the excitation spectrum and show that it is most convenient to regard it as a double spectrum---not one as is ordinarily supposed.
We propose an exactly solvable model of one-dimensional anyons with competing δ-function and derivative δ-function interaction potentials. The Bethe ansatz equations are derived in terms of the N -particle sector for the quantum anyonic field model of the generalized derivative nonlinear Schrödinger equation. This more general anyon model exhibits richer physics than that of the recently studied one-dimensional model of δ-function interacting anyons. We show that the anyonic signature is inextricably related to the velocities of the colliding particles and the pairwise dynamical interaction between particles.
In order to analyze a limiting case of the 1D delta anyon model, the coupling strength of the δ interaction c is modified to become a function of the anyonic parameter κ. A pedagogic derivation of the solution for this modified model using the method of anyon-boson mapping plus Bethe ansatz is presented. The limiting case as κ → π and simultaneously c → 0, which was previously neglected, is analyzed. Some unexpected properties of this limiting case are discovered. The BAEs are compared with previous results.
We show that below the degenerate temperature the distribution profiles of strongly interacting anyons in one-dimension coincide with the most probable distributions of ideal particles obeying generalized exclusion statistics (GES). In the strongly interacting regime the thermodynamics and the local two-particle correlation function derived from the GES are seen to agree for low temperatures with the results derived for the anyon model using the thermodynamic Bethe ansatz. The anyonic and dynamical interactions implement a continuous range of GES, providing a signature of strongly interacting anyons, including the strongly interacting one-dimensional Bose gas.