Quantum integrability in the multistate Landau-Zener problem

TL;DR

This paper investigates the conditions under which multistate Landau-Zener problems are exactly solvable, revealing that they are either reducible to 2x2 cases or belong to special families of commuting Hamiltonians, thus linking integrability to solvability.

Contribution

It classifies Hamiltonians with exact solutions in the multistate Landau-Zener problem and constructs explicit commuting families, unveiling new integrable models and conjecturing integrability as a solvability criterion.

Findings

Equal slope model is part of a maximal commuting family linear in time.

Bow-tie model belongs to a maximal quadratic commuting family.

Solutions to various Landau-Zener problems are derived from the 2x2 case.

Abstract

We analyze Hamiltonians linear in the time variable for which the multistate Landau-Zener problem is known to have an exact solution. We show that they either belong to families of mutually commuting Hamiltonians polynomial in time or reduce to the 2 x 2 Landau-Zener problem, which is considered trivially integrable. The former category includes the equal slope, bow-tie, and generalized bow-tie models. For each of these models we explicitly construct the corresponding families of commuting matrices. The equal slope model is a member of an integrable family that consists of the maximum possible number (for a given matrix size) of commuting matrices linear in time. The bow-tie model belongs to a previously unknown, similarly maximal family of quadratic commuting matrices. We thus conjecture that quantum integrability understood as the existence of nontrivial parameter-dependent commuting partners is a necessary condition for the Landau-Zener solvability. Descendants of the 2 x 2 Landau-Zener Hamiltonian are e.g. general SU(2) and SU(1,1) Hamiltonians, time-dependent linear chain, linear, nonlinear, and double oscillators. We explicitly obtain solutions to all these Landau-Zener problems from the 2 x 2 case.