[Paper Review] A discrete analogue of periodic delta Bose gas and affine Hecke algebra

Yoshihiro Takeyama|arXiv (Cornell University)|Sep 13, 2012

Cold Atom Physics and Bose-Einstein Condensates14 references3 citations

TL;DR

This paper introduces a two-parameter discrete analogue of the periodic delta Bose gas Hamiltonian on a lattice, constructed via integral-reflection operators that realize a representation of the affine Hecke algebra. It establishes a propagation operator linking eigenfunctions of a 'half-Laplacian' to those of the full Hamiltonian, and constructs symmetric, periodic eigenfunctions (Bethe wave functions) via the Bethe ansatz; in the special case α=0, these reduce to specializations of Hall-Littlewood polynomials.

ABSTRACT

We consider an eigenvalue problem for a discrete analogue of the Hamiltonian of the non-ideal Bose gas with delta-potentials on a circle. It is a two-parameter deformation of the discrete Hamiltonian for joint moments of the partition function of the O'Connell-Yor semi-discrete polymer. We construct the propagation operator by using integral-reflection operators, which give a representation of the affine Hecke algebra. We also construct eigenfunctions by means of the Bethe ansatz method.

Motivation & Objective

• To develop a discrete, two-parameter deformation of the periodic delta Bose gas Hamiltonian on a lattice.
• To construct a propagation operator using integral-reflection operators that represent the affine Hecke algebra.
• To solve the eigenvalue problem for the Hamiltonian using the Bethe ansatz method.
• To identify conditions under which the eigenfunctions reduce to known special functions, specifically Hall-Littlewood polynomials.
• To establish a connection between integrable stochastic models and algebraic structures such as the affine Hecke algebra.

Proposed method

• The Hamiltonian H is defined on the integer lattice X = ⊕ℤv_i using shift operators t_{v_i} and functions d_i^± that count affine root system conditions.
• Integral-reflection operators Q_i are constructed from divided difference operators satisfying braid relations, forming a representation of the affine Hecke algebra.
• A propagation operator G is defined such that it maps eigenfunctions of ∑t_{v_i} to eigenfunctions of H with the same eigenvalue.
• The Bethe ansatz method is applied to construct symmetric, periodic eigenfunctions (Bethe wave functions) by solving a system of algebraic equations for spectral parameters p_i.
• The construction relies on the commutation relations between shift operators and integral-reflection operators, particularly t_{v_j+1}Q_j = Q_j t_{v_j} + α + (1−β)t_{v_j+1}.
• In the α=0 case, the eigenfunctions are shown to be expressible as products involving Hall-Littlewood polynomials via specialization of variables.

Experimental results

Research questions

• RQ1 How can a discrete analogue of the periodic delta Bose gas Hamiltonian be formulated with two deformation parameters?
• RQ2 What algebraic structure underlies the propagation operator connecting eigenfunctions of the half-Laplacian to those of the full Hamiltonian?
• RQ3 How do the eigenfunctions of the discrete Hamiltonian relate to known special functions such as Hall-Littlewood polynomials?
• RQ4 What role do integral-reflection operators play in realizing representations of the affine Hecke algebra in this discrete setting?
• RQ5 Under what conditions does the Bethe ansatz yield symmetric and periodic eigenfunctions for the lattice Hamiltonian?

Key findings

• The propagation operator G maps eigenfunctions of ∑_{i=1}^k t_{v_i} to eigenfunctions of H with the same eigenvalue, establishing a key link between simpler and full Hamiltonians.
• The eigenfunctions constructed via the Bethe ansatz are symmetric and periodic under the action of the extended Weyl group Ŵ, ensuring compatibility with periodic boundary conditions.
• When α=0, the Bethe wave functions h_p(x) are shown to be proportional to Hall-Littlewood polynomials via the expression Δ(p)R_ε(x)(p₁⁻¹,…,p_k⁻¹;β).
• The system of algebraic equations (6.1) for spectral parameters p_i ensures that the Bethe wave functions are eigenfunctions of H with eigenvalue ∑_{i=1}^k p_i.
• The integral-reflection operators Q_i provide a representation of the affine Hecke algebra of type GL_k, generalizing earlier constructions from polynomial representations.
• The construction is consistent under the action of the affine Weyl group, with G(f)(πx) = G(f)(x), ensuring periodicity of the resulting eigenfunctions.