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International Journal of Modern Physics A

STATISTICS IN ONE DIMENSION

Charilaos Aneziris, A. P. Balachandran, Diptiman Sen

Published Nov 10, 1991• 32 citations
Publisher

Abstract

We show that there are novel generalizations of statistics for identical particles in one dimension. These arise due to possible boundary conditions on wave functions, or equivalently due to δ-function interactions, at points of coincidence of particle coordinates. Special choices of these boundary conditions describe bosons, fermions or paraparticles. The general solution for the boundary conditions involves vector-valued wave functions and statistics with non-Abelian features, even though the classical configuration space has an Abelian fundamental group. Physical models leading to such non-Abelian statistics, involving for example δ-function interactions of spins, are constructed. Properties under parity and time reversal of the new boundary conditions are studied. It is shown that the Bethe ansatz does not always give eigenstates of energy. When it does, the wave numbers in the ansatz for three or more particles must satisfy a weaker form of the Yang-Baxter equations and certain additional equations. The energy spectrum and wave functions for identical particles on R1 or S1 are discussed in some simple cases. Our work generalizes the previous work of Lieb and Liniger, Lieb, Yang and Leinaas and Myrheim.

DOI
10.1142/s0217751x91002240

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We show that there are novel generalizations of statistics for identical particles in one dimension. These arise due to possible boundary conditions on wave functions, or equivalently due to δ-function interactions, at points of coincidence of particle coordinates. Special choices of these boundary conditions describe bosons, fermions or paraparticles. The general solution for the boundary conditions involves vector-valued wave functions and statistics with non-Abelian features, even though the classical configuration space has an Abelian fundamental group. Physical models leading to such non-Abelian statistics, involving for example δ-function interactions of spins, are constructed. Properties under parity and time reversal of the new boundary conditions are studied. It is shown that the Bethe ansatz does not always give eigenstates of energy. When it does, the wave numbers in the ansatz for three or more particles must satisfy a weaker form of the Yang-Baxter equations and certain additional equations. The energy spectrum and wave functions for identical particles on R1 or S1 are discussed in some simple cases. Our work generalizes the previous work of Lieb and Liniger, Lieb, Yang and Leinaas and Myrheim.
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.
We show that the thermodynamic Bethe ansatz equations for one-dimensional integrable many-body systems can be reinterpreted in such a way that they only code the statistical interactions, in the sense of Haldane, between particles of identical or different momenta. Thus, the thermodynamic properties of these systems can be characterized by the generalized ideal gases recently proposed by one of us. For example, the Yang-Yang δδδ-function gas is a gas with specific statistical interactions between particles of different momenta, while the Calogero-Sutherland system provides a model for an ideal gas of particles with a fractional statistics.
As is well known, there exists a four-parameter family of local interactions in 1D. We interpret these parameters as coupling constants of delta-type interactions which include different kinds of momentum-dependent terms, and determine all cases leading to many-body systems of distinguishable particles which are exactly solvable by the coordinate Bethe ansatz. We find two such families of systems, one with two independent coupling constants deforming the well-known delta-interaction model to non-identical particles, and the other with a particular one-parameter combination of the delta and (the so-called) delta-prime interaction. We also find that the model of non-identical particles gives rise to a somewhat unusual solution of the Yang–Baxter relations. For the other model we write down explicit formulae for all eigenfunctions.
As is well known, there exists a four-parameter family of local interactions in 1D. We interpret these parameters as coupling constants of delta-type interactions which include different kinds of momentum-dependent terms, and determine all cases leading to many-body systems of distinguishable particles which are exactly solvable by the coordinate Bethe ansatz. We find two such families of systems, one with two independent coupling constants deforming the well-known delta-interaction model to non-identical particles, and the other with a particular one-parameter combination of the delta and (the so-called) delta-prime interaction. We also find that the model of non-identical particles gives rise to a somewhat unusual solution of the Yang–Baxter relations. For the other model we write down explicit formulae for all eigenfunctions.
Introduction The one-dimensional Bose gas with point-like interaction of the particles (the quantum variant of the nonlinear Schrödinger equation) is one of the main and most important models which can be Solved by means of the Bethe Ansatz. This model has been thoroughly investigated (and. We shall start with the construction of eigenfunctions of the Hamiltonian in a finite volume. Quantities interesting from the physical point of view (in the thermodynamic limit at zero temperature) are considered; the thermodynamics at finite temperatures is investigated in detail as well. A number of essential ideas which will be applied to other models are introduced. The construction of eigenfunctions of the Hamiltonian is explained in section 1. Their explicit form and, in particular, the two-particle reducibility, are common features of the models solvable by means of the Bethe Ansatz method. Periodic boundary conditions are imposed on the wave function in section 2; the Bethe equations for the particle' momenta are introduced and analyzed. Taken in the logarithmic form, these equations realize the extremum condition of a certain functional, the corresponding action being called the Yang-Yang action. The transition to the thermodynamic limit is considered in section 3. In that same section the ground state of the gas is constructed. The density of the particle distribution in momentum space and the energy of the ground state are calculated. The method of the transition to the thermodynamic limit described in this section is rather general and may be applied to any model solvable by means of the Bethe Ansatz. In section 4, excitations above the ground state are constructed and their main characteristics (energy, momentum and scattering matrix) are determined by means of the dressing equations. The ground state of the model is the Dirac sea (also called the Fermi sphere).
The repulsive delta-function interaction model in one dimension is reviewed for spinless particles and for spin-1/2 fermions. The problem of solving the differential equation related to the Schrödinger equation is reduced by the Bethe ansatz to a system of algebraic equations. The delta-function interaction is shown to have no effect on spinless fermions which therefore behave like free fermions, in agreement with Pauli's exclusion principle. The ground-state problem of spinless bosons is reduced to an inhomogeneous Fredholm equation of the second kind. In the limit of impenetrable interactions, the spinless bosons are shown to have the energy spectrum of free fermions. The model for spin-1/2 fermions is reduced by the Bethe ansatz to an eigenvalue problem of matrices of the same sizes as the irreducible representations R of the permutation group of N elements. For some R's this eigenvalue problem itself is solved by a generalized Bethe ansatz. The ground-state problem of spin-1/2 fermions is reduced to a generalized Fredholm equation.