Exact Solution of DoubleδFunction Bose Gas through an Interacting Anyon Gas

Abstract

1d Bose gas interacting through δ, δ ′ and double-δ function potentials is shown to be equivalent to a δ anyon gas allowing exact Bethe ansatz solution. In the noninteracting limit it describes an ideal gas with generalized exclusion statistics and solves some recent controversies.PACS numbers: 05.30.Jp, 03.70.+k 11.55.Ds, 71.10.Pm, The concept of particles with generalized exclusion statistics (GES) introduced by Haldane [1] has important consequences [2] in describing 1d non Fermi-liquids [3], which in turn is believed to be related [4] to the edge excitations in fractional quantum Hall effect [5]. On the other hand, inspired by the success of the Chern-Simon theory, an attempt was made recently [6] to describe a 1d ideal gas with GES in the framework of a gauge field model. However, in a subsequent paper [7] the previous result was shown to be wrong and some other conclusion was offered. Our aim here is to deal primarily with a 1d Bose gas interacting through double-δ function potentials together with the well known δ and derivative δ-function interactions. We show that this interacting model with several singular potentials is equivalent to a 1d gas with GES (which we call anyon for brevity) interacting via δ-function potential only. This δ-anyon gas is found to be exactly solvable by the coordinate Bethe ansatz (CBA) just like its bosonic counterpart, contradicting the common belief [8] that the CBA is applicable only to models with symmetric or antisymmetric wave functions. Remarkably, at the limit of vanishing interaction the anyon gas becomes free and gauge equivalent to a related model proposed in [6]. This shows that, though the explicit wave function and the N -body Hamiltonian of [6] are not exact, the conclusion it arrived at is basically correct. Therefore, while the error in the treatment of [6] was detected in [7], the source of this error and the possible way to rectify it becomes evident from our result.We start with a 1d Bose gas interacting through generalized δ function potentials as(1)This model was briefly considered and readily discarded in [9] as too difficult a problem to solve. Notice however that for γ a = 0, a = 1, 2, i.e. without the double-δ potentials, it has various exactly solvable limits. For example, for κ = 0, c = 0 the model becomes the well known δ-Bose gas [10], while for κ = 0, c = 0 it corresponds to Bose gas with δ ′ interaction [11]. Both these cases are not only exactly solvable by CBA, but also represent quantum integrable systems allowing R-matrix solution. This can be proved through their connection with the quantum integrable nonlinear Schrödinger equation (NLSE) [12] and derivative NLSE [13], respectively. Even the mixed case with κ = 0, c = 0 is solvable through CBA [8,11], though as a quantum model it does not allow a R-matrix solution. Nevertheless for γ a = 0, i.e. with the inclusion of highly singular double-δ function interactions, the solvability of the model is completely lost and the application of the CBA becomes problematic due to the presence of tree-b...

Junctions of anyonic Luttinger wires

“…Thus, when κ = 1 the model is the well-known fermionic TL model, while the bosonic limit κ → 0 is not well defined in this formalism as will be clearer in the following. We stress that this anyonic model is different from the gases discussed elsewhere 42,43,46,49,56 , that also have a Luttinger liquid description. As in the fermionic case, the model is naturally solved exactly through bosonization 27 .…”

“…The reason for this generalization is twofold. On one hand the study of 1D anyonic model is attracting a renewed interest [42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59] , mainly motivated by possible experiments with cold atoms 60 . On the other hand, the transport of wires joined with a quantum Hall island is driven by anyonic excitations 12 .…”

One-Dimensional Interacting Anyon Gas: Low-Energy Properties and Haldane Exclusion Statistics

“…In the sense of Haldane exclusion statistics, the 1D interacting Bose gas is equivalent to the ideal gas with generalized fractional statistics [8,9]. We consider an integrable model of anyons with a -function interaction introduced and solved by Kundu [10]. Here we obtain the low-energy properties and Haldane exclusion statistics of this 1D anyon gas.…”

Dynamical properties of hard-core anyons in one-dimensional optical lattices

“…[13], 1D anyon gas has been investigated theoretically in various 1D systems [19][20][21][22] including the Bose quantum gas. Kundu proved that a 1D Bose gas interacting through δ-function potential combined with double δ-function potential and derivative δ-function potential is equivalent to the anyon gas interacting via δ-function potential [21]. This stimulated many research interests on δ-anyon gas [22][23][24][25][26][27][28][29][30][31][32].…”