Integrable N ≤ O Component Nonlinear Schrödinger Model, Phase Transitions and Supersymmetry

Abstract

Supersymmetry concepts originally introduced in the high-energy physics have recently penetrated in the nuclear physics, condensed matter and statistical mechanics [1], In the last cases anticommuting variables are first introduced for pure combinatorical reasons, to replace the N→0 replica trick. But recently there have been made an attempt to describe the High Temperature Superconductivity (HTSC) and strong correlated electronic systems in general, using symmetry between fermionic and bosonic excitations (hole and spin excitations) [2]. The order parameter in this case can appear in a very sophisticated form as the Grassman constant or the Clifford number. Related to this integrable generalizations of the Ginzburg-Landau model are possible in the form of the vector or matrix Nonlinear Schrödinger Equations (NLSE) with global symmetry in the space of order parameters [3]. The close analogies between antiferromagnetism and superfluidity in the context of classical integrable models have exact meaning in the form of the gauge-equivalence of the Heisenberg model and NLSE.