Parameter Dependent Commuting Matrices, Pl\"ucker relations and Related Quantum Glass Models

Abstract

Type-I matrices were introduced recently as finite dimensional prototypes of quantum integrable systems. These matrices are linearly dependent on an "interaction" type parameter, and possess interesting properties such as commuting partner matrices and generically violate the von Neumann Wigner non crossing rule. The important role of Pl\"ucker relations in this construction is noted. Type-I matrices are given a transparent formulation in terms of Fermi or Bose type particle operators- they represent a Quantum glass model with either Fermi or Bose statistics, with several free parameters that may be chosen at will.