Knizhnik-Zamolodchikov equations and integrable hyperbolic Landau-Zener models

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.