Knizhnik-Zamolodchikov equations and integrable Landau-Zener models

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 the probabilities of tunneling from eigenstates at $t=t_\text{in}$ to the 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 the tunneling probabilities are elementary functions of Hamiltonian parameters and time-dependent wavefunctions are special functions. It has recently been proposed that exactly solvable LZ models map to KZ equations. Here we use this connection to identify and solve various integrable hyperbolic LZ models $\hat H(t)=\hat A+\hat B/t$ for $N=2, 3$, and $4$, where $\hat A$ and $\hat B$ are time-independent matrices. Some of these models have been considered, though not fully solved, before and others are entirely new.
Comment: 34 pages, 8 figures