Operator Algebra Approach to Soluble Models of Quantum Spin Lattice Systems

A. Crubellier, A. Janner, Ted Janssen

H. Y. Shik, Y.Q. Li, H. Q. Lin

Murray T. Batchelor, Michael N. Barber

The authors construct a family of isotropic spin-s quantum chains consisting of sums of operators satisfying a Temperley-Lieb algebra. Exact values for the infinite lattice limit of the ground-state energy per site and for the (non-zero) gap to the lowest energy excited state follow from the Temperley-Lieb equivalence with a Bethe ansatz soluble XXZ model. The family of spin chains includes the biquadratic spin-1 model.

Pedro de M. Rios, Eldar Straume

Michael Levin, Xiao-Gang Wen

We investigate the general properties of lattice spin models with emerging fermionic excitations. We argue that fermions always come in pairs and their creation operator always has a stringlike structure with the newly created particles appearing at the end points of the string. The physical implication of this structure is that the fermions always couple to a nontrivial gauge field. We present exactly soluble examples of this phenomenon in two and three dimensions. Our analysis is based on an algebraic formula that relates the statistics of a lattice particle to the properties of its hopping operators. This approach has the advantage in that it works in any number of dimensions---unlike the flux-binding picture developed in fractional quantum Hall theory.

T. D. Schultz, Daniel Charles Mattis, Élliott H. Lieb

The two-dimensional Ising model for a system of interacting spins (or for the ordering of an AB alloy) on a square lattice is one of the very few nontrivial many-body problems that is exactly soluble and shows a phase transition. Although the exact solution in the absence of an external magnetic field was first given almost twenty years ago in a famous paper by Onsager1 using the theory of Lie algebras, the flow of papers on both approximate and exact methods has remained strong to this day.2 One reason for this has been the interest in testing approximate methods on an exactly soluble problem. A second reason, no doubt, has been the considerable formidability of the Onsager method. The simplification achieved by Bruria Kaufman3 using the theory of spinor representations has diminished, but not removed, the reputation of the Onsager approach for incomprehensibility, while the subsequent application of this method by Yang4 to the calculation of the spontaneous magnetization has, if anything, helped to restore this reputation.