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

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.

41 pages, 4 figures, 1 table

References in corpus (2)

• Level crossing in the three-body problem for strongly interacting fermions in a harmonic trap
• The Link between Integrability, Level Crossings, and Exact Solution in Quantum Models

Cited by in corpus (19)

• Integrable time-dependent quantum Hamiltonians
• Integrals of motion for one-dimensional Anderson localized systems
• Quantum integrability in systems with finite number of levels
• Proof of the absence of local conserved quantities in the XYZ chain with a magnetic field
• Integrable time-dependent Hamiltonians, solvable Landau-Zener models and Gaudin magnets
• Quantum integrability in the multistate Landau-Zener problem
• Integrable matrix theory: Level statistics
• Generalized microcanonical and Gibbs ensembles in classical and quantum integrable dynamics
• Absence of local conserved quantity in the Heisenberg model with next-nearest-neighbor interaction
• Detuning-induced robustness of a three-state Lanzau-Zener model against dissipation
• Three-state Landau-Zener model in the presence of dissipation
• A remark on the notion of independence of quantum integrals of motion in the thermodynamic limit
• Rotationally invariant ensembles of integrable matrices
• Master equation approach to the three-state open Majorana model
• Functionally independent conservations laws in a quantum integrable model
• Knizhnik-Zamolodchikov equations and integrable hyperbolic Landau-Zener models
• Resonant transitions due to changing boundaries
• Open multistate Majorana model
• Weyl's Relations, Integrable Matrix Models and Quantum Computation