Elliptic Calogero-Sutherland Research Articles

How Coordinate Bethe Ansatz Works for Inozemtsev Model

Three decades ago, Inozemtsev found an isotropic long-range spin chain with elliptic pair potential that interpolates between the Heisenberg and Haldane-Shastry (HS) spin chains while admitting an exact solution throughout, based on a connection with the elliptic quantum Calogero-Sutherland model. Though Inozemtsev's spin chain is widely believed to be quantum integrable, the underlying algebraic reason for its exact solvability is not yet well understood. As a step in this direction we refine Inozemtsev's `extended coordinate Bethe ansatz' and clarify various aspects of the model's exact spectrum and its limits. We identify quasimomenta in terms of which the $M$-particle energy is close to being (functionally) additive, as one would expect from the limiting models; our expression is additive iff the energy of the elliptic Calogero-Sutherland system is so. This enables us to rewrite the energy and Bethe-ansatz equations on the elliptic curve, turning the spectral problem into a rational problem as might be expected for an isotropic spin chain. We treat the $M=2$ particle sector and its limits in detail. We identify an $S$-matrix that is independent of positions. We show that the Bethe-ansatz equations reduce to those of Heisenberg in one limit and give rise to the `motifs' of HS in the other limit. We show that, as the interpolation parameter changes, the `scattering states' from Heisenberg become Yangian highest-weight states for HS, while bound states become ($\mathfrak{sl}_2$-highest weight versions of) affine descendants of the magnons from $M=1$. For bound states we find a generalisation of the known equation for the `critical length' for the Heisenberg spin chain. We discuss completeness for $M=2$ by passing to the elliptic curve. Our review of the two-particle sectors of the Heisenberg and HS spin chains may be of independent interest.

Explicit Solution of the (Quantum) Elliptic Calogero–Sutherland Model

We derive explicit formulas for the eigenfunctions and eigenvalues of the elliptic Calogero-Sutherland model as infinite series, to all orders and for arbitrary particle numbers and coupling parameters. The eigenfunctions obtained provide an elliptic deformation of the Jack polynomials. We prove in certain special cases that these series have a finite radius of convergence in the nome $q$ of the elliptic functions, including the two particle (= Lam\'e) case for non-integer coupling parameters.

Remarkable identities related to the (quantum) elliptic Calogero-Sutherland model

We present remarkable functional identities related to the elliptic Calogero-Sutherland (eCS) system. We derive them from a second quantization of the eCS model within a quantum field theory model of anyons on a circle and at finite temperature. The identities involve two eCS Hamiltonians with arbitrary and, in general, different particle numbers N and M, and a particular function of N+M variables arising as anyon correlation function of N particles and M antiparticles. In addition to identities obtained from anyons with the same statistics parameter λ, we also obtain “dual” relations involving “mixed” correlation functions of anyons with two different statistics parameters λ and 1∕λ. We also give alternative, elementary proofs of these identities by direct computations.

Second Quantization of the Elliptic Calogero-Sutherland Model

We construct a quantum field theory model of anyons on a circle and at finite temperature. We find an anyon Hamiltonian providing a second quantization of the elliptic Calogero-Sutherland model. This allows us to prove a remarkable identity which is a starting point for an algorithm to construct eigenfunctions and eigenvalues of the elliptic Calogero-Sutherland Hamiltonian.

Anyons and the elliptic Calogero-Sutherland model

We obtain a second quantization of the elliptic Calogero-Sutherland (eCS) model by constructing a quantum field theory model of anyons on a circle and at a finite temperature. This yields a remarkable identity involving anyon correlation functions and providing an algorithm for solving of the eCS model. The eigenfunctions obtained define an elliptic generalization of the Jack polynomials.