Classification of Parameter-Dependent Quantum Integrable Models, Their Parameterization, Exact Solution, and Other Properties

TL;DR

This paper classifies quantum integrable models linear in a coupling constant, provides exact solutions for their spectra, and explores their properties and classifications, including applications to the Hubbard model.

Contribution

It introduces a new classification scheme for parameter-dependent quantum integrable models and offers an exact parameterization and solution for these models, covering all Type 1, 2, and 3 cases.

Findings

Parameterization covers all Type 1, 2, and 3 models.

Exact eigenvalues and eigenvectors are obtained.

Numerical analysis of energy level crossings and model taxonomy.

Abstract

We study general quantum integrable Hamiltonians linear in a coupling constant and represented by finite NxN real symmetric matrices. The restriction on the coupling dependence leads to a natural notion of nontrivial integrals of motion and classification of integrable families into Types according to the number of such integrals. A Type M family in our definition is formed by N-M nontrivial mutually commuting operators linear in the coupling. Working from this definition alone, we parameterize Type M operators, i.e. resolve the commutation relations, and obtain an exact solution for their eigenvalues and eigenvectors. We show that our parameterization covers all Type 1, 2, and 3 integrable models and discuss the extent to which it is complete for other types. We also present robust numerical observation on the number of energy level crossings in Type M integrable systems and analyze the taxonomy of types in the 1d Hubbard model.