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

TL;DR

This paper explores parameter-dependent Type-I matrices, their connection to Pl"ucker relations, and their realization as quantum glass models with Fermi or Bose particles, revealing new integrability properties.

Contribution

It introduces a transparent formulation of Type-I matrices using particle operators and links them to quantum glass models, highlighting their algebraic and physical properties.

Findings

Type-I matrices depend on parameters and commute with partners.

They violate the von Neumann Wigner non crossing rule.

They can be represented as quantum glass models with Fermi or Bose statistics.

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.