New family of solvable 1D Heisenberg models

Abstract

Starting from a Calogero-Sutherland model with hyperbolic interaction confined by an external field with Morse potential we construct a Heisenberg spin chain with exchange interaction ∝ 1/ sinh 2 x on a lattice given in terms of the zeroes 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 nonuniform nearest neighbour coupling in a nonuniform magnetic field which allows to study the thermodynamics in the limit of infinite chains.

Thermodynamics of spin chains of Haldane–Shastry type and one-dimensional vertex models

“…The main idea behind our approach is to exploit the equivalence of these chains with a corresponding family of Ising-like inhomogeneous vertex models. In the absence of an external magnetic field, this fact had already been conjectured by Frahm [5] and by Frahm and Inozemtsev [6], and was recently established in a rigorous way in Ref. [35].…”

Специальное Множество Собственных Векторов Для Гиперболических Систем Сазерленда

“…Что касается их спектральных свойств, то общим является наличие огромных вырождений уровней; эти вырождения были успешно классифицирова-ны с использованием внутренней симметрии янговского типа [13], [19]. Для более сложных систем типа СМ спектральная задача была решена только для семейства неравномерных спиновых решеток [21].…”

“…С учетом результатов работ [11]- [21] возникает естественный вопрос, существует ли решаемая спиновая версия динамических систем СМ, описываемых гамильто-нианом…”

A special set of eigenvectors for the hyperbolic Sutherland systems

“…The common feature of their spectral properties is the presence of huge level degeneracies, which were successfully classified using the internal symmetry of Yangian type [13], [19]. For more complicated SMtype systems, the spectral problem has been solved only for the family of nonuniform spin lattices [21].…”

“…In view of the results in [11]- [21], a natural question is whether there exists a solvable spin version of the dynamical SM systems described by the Hamiltonian…”