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Theoretical and Mathematical Physics

Algebraic approach to the solution of a one-dimensional model of N interacting particles

A. M. Perelomov

Published Mar 1, 1971• 55 citations
Publisher
DOI
10.1007/bf01030108

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A high temperature expansion is employed to map some complex anisotropic nonhermitian three dimensional Ising models with algebraic long range interactions into a solvable two dimensional variant. Some solutions are presented. This framework further allows for some very simple general observations. It will be seen that the absence of a “phase interference ” effect plays an important role in high dimensional problems. A very forbidding purely algebraic recursive series solution to the three dimensional nearest neighbor Ising model will be given. In the aftermath, the full-blown three dimensional nearest neighbor Ising model is exactly mapped onto a single spin 1/2 particle with nontrivial dynamics. I.
One-dimensional quantum particle system with S U (ν) spins interacting through inverse square interactions is studied. We reveal algebraic structures of the system: hidden symmetry is the U(ν)≅SU(ν)⊗U(1)U(\nu )\cong SU(\nu )\otimes U(1)U(ν)≅SU(ν)⊗U(1) current algebra. This is consistent with the fact that the ground state wave function is a solution of the Knizhnik-Zamolodchikov equation. Furthermore we show that the system has a higher symmetry, which is the w 1+∞ algebra. With this W algebra we can clarify simultaneously the structures of the Calogero type (1/ x 2 -interactions) and Sutherland type (1/sin 2 x -interactions). The Yangian symmetry is briefly discussed.