Knizhnik-Zamolodchikov equations and integrable hyperbolic Landau-Zener models

TL;DR

This paper explores the connection between integrable Landau-Zamner models with hyperbolic time dependence and Knizhnik-Zamolodchikov equations, providing exact solutions for specific low-dimensional cases.

Contribution

It identifies and solves a class of integrable hyperbolic Landau-Zamolodchikov models using their relationship with KZ equations for low-dimensional systems.

Findings

Exact solutions for N=2, 3, 4 hyperbolic LZ models

Established link between LZ models and KZ equations

Demonstrated integrability of specific hyperbolic LZ Hamiltonians

Abstract

We study the relationship between integrable Landau-Zener (LZ) models and Knizhnik-Zamolodchikov (KZ) equations. The latter are originally equations for the correlation functions of two-dimensional conformal field theories, but can also be interpreted as multi-time Schr\"odinger equations. The general LZ problem is to find probabilities of tunneling from eigenstates at $t=t_\text{in}$ to eigenstates at $t\to+\infty$ for an $N\times N$ time-dependent Hamiltonian $\hat H(t)$. A number of such problems are exactly solvable in the sense that their tunneling probabilities are elementary functions of Hamiltonian parameters. Recently, it has been proposed that exactly solvable LZ models of this type map to KZ equations. Here we use this connection to identify and solve a class of integrable LZ models with hyperbolic time dependence, $\hat H(t)=\hat A+\hat B/t$, for $N=2, 3$, and $4$, where $\hat A$ and $\hat B$ are time-independent matrices.