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arXiv:nlin/0611008 · Symmetry Integrability and Geometry Methods and Applications

Quasi-Exactly Solvable N-Body Spin Hamiltonians with Short-Range Interaction Potentials

Alberto Enciso B

Published Nov 3, 2006• 1 citation
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Abstract

We review some recent results on quasi-exactly solvable spin models presenting near-neighbors interactions.These systems can be understood as cyclic generalizations of the usual Calogero-Sutherland models.A nontrivial modification of the exchange operator formalism is used to obtain several infinite families of eigenfunctions of these models in closed form.

DOI
10.3842/sigma.2006.073

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We review some recent results on quasi-exactly solvable spin models presenting near-neighbors interactions.These systems can be understood as cyclic generalizations of the usual Calogero-Sutherland models.A nontrivial modification of the exchange operator formalism is used to obtain several infinite families of eigenfunctions of these models in closed form.
We present a detailed analysis of the spin models with near-neighbours interactions constructed in our previous paper (Enciso et al 2005 Phys. Lett. B 605 214) by a suitable generalization of the exchange operator formalism. We provide a complete description of a certain flag of finite-dimensional spaces of spin functions preserved by the Hamiltonian of each model. By explicitly diagonalizing the Hamiltonian in the latter spaces, we compute several infinite families of eigenfunctions of the above models in closed form in terms of generalized Laguerre and Jacobi polynomials.
The bound and scattering states of the many-body systems, related to the short-range Dyson model, are studied. The Hamiltonian for the full-line problem is connected to decoupled oscillators. The analytically obtainable eigenstates are smaller in number as compared to the Calogero-Sutherland family, indicating the quasiexactly solvable nature of these models. The exactly found scattering states, a smaller set as compared to the Calogero case, can be realized as coherent states. The relation of the scattering Hamiltonian to free particles is also established algebraically. We analyze both ${A}_{N\ensuremath{-}1}$ and BCNB{C}_{N}BCN​ models on a circle and construct a part of the excitation spectrum by making use of the symmetry arguments.
Starting from a Calogero-Sutherland model with the hyperbolic interaction confined by an external field with Morse potential, we construct a Heisenberg spin chain with exchange interaction varies as 1/sinh 2 x on a lattice given in terms of the zeros of Laguerre polynomials. Varying the strength of the Morse potential, the Haldane-Shastry and harmonic spin chains are reproduced. The spectrum of the models in this class is found to be that of a classical one-dimensional Ising chain with non-uniform nearest-neighbour coupling in a non-uniform magnetic field which allows us to study the thermodynamics in the limit of infinite chains.
We introduce a generalized Sutherland model: a system of one-dimensional fermions or bosons interacting by a two-body potential which has a short-range interaction of finite range and amplitude, and an inverse-square tail. Using renromalization-group arguments we analyze the ground-state and correlation properties of this family of models. We reproduce known exact results as special cases of the general model and point out a correspondence between the exactly solvable problems and special initial or final conditions of the renormalization transformation, which we believe to be a general criterion. This enables us to predict a series of new presumably exact results.
This thesis is devoted to the study of various examples of exactly solved quantum many-body systems in one-dimension. It is divided into two parts: the first provides background and complementary results to the second, which consists of three scientific papers. The first paper concerns a particu- lar extension, corresponding to the root system CN, of the delta-interaction model. We prove by construction that its exact solution, even in the gen- eral case of distinguishable particles, can be obtained by the coordinate Bethe ansatz. We also elaborate on the physical interpretation of this model. It is well known that the delta-interaction is included in a four parameter family of local interactions. In the second paper we interpret these parameters as cou- pling constants of certain momentum dependent interactions and determine all cases leading to a many-body system of distinguishable particles which can be solved by the coordinate Bethe ansatz. In the third paper we consider the so-called rational Calogero-Sutherland model, describing an arbitrary number of particles on the real line, confined by a harmonic oscillator potential and interacting via a two-body interaction proportional to the inverse square of the inter-particle distance. We construct a novel solution algorithm for this model which enables us to obtain explicit formulas for its eigenfunctions. We also show that our algorithm applies, with minor changes, to all extensions of the rational Calogero-Sutherland model which correspond to a classical root system.
It is shown that the description of the states of infinite S = 1/2 interacting spin systems with the Hamiltonian H s J Z a2 sinh-Za(J / ) (~ 1) 2 j# t , j , l~Z 2 can be performed by studying the hyperbolic Calogero-Sutherland eigenvalue problem. The construction of multimagnon wave functions in each N-magnon sector is based on solutions of the set of linear algebraic equations which determine also the structure of zonal spherical functions on symmetric spaces X ~ = SL(N, H)/Sp(N) of negative curvature. The usual Bethe Ansatz for the XXX Heisenberg model corresponds to asymptotic forms of these wave functions at small values of a 1 or large distances between spins turned over the ferromagnetic ground state.