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 Schrödinger equations. The general LZ problem is to find probabilities of tunneling from eigenstates at t=t_\text{in} t = t in to eigenstates at t\to+∞ t → + ∞ for an N× N N × N time-dependent Hamiltonian \hat H(t) 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 H ̂ ( t ) = A ̂ + B ̂ / t , for N=2, 3 N = 2 , 3 , and 4 4 , where \hat A A ̂ and \hat B B ̂ are time-independent matrices.
Abstract Recently, integrability conditions (ICs) in mutistate Landau-Zener (MLZ) theory were proposed [1]. They describe common properties of all known solved systems with linearly time-dependent Hamiltonians. Here we show that ICs enable efficient computer assisted search for new solvable MLZ models that span complexity range from several interacting states to mesoscopic systems with many-body dynamics and combinatorially large phase space. This diversity suggests that nontrivial solvable MLZ models are numerous. In addition, we refine the formulation of ICs and extend the class of solvable systems to models with points of multiple diabatic level crossing.
The search for exactly solvable models is an evergreen topic in theoretical physics. In the context of multistate Landau-Zener models --
N-state quantum systems with linearly time-dependent Hamiltonians -- the theory of integrability provides a framework for identifying new solvable cases. In particular, it was proved that the integrability of a specific class known as the multitime Landau-Zener (MTLZ) models guarantees their exact solvability. A key finding was that an
N-state MTLZ model can be represented by data defined on an
N-vertex graph. While known host graphs for MTLZ models include hypercubes, fans, and their Cartesian products, no other families have been discovered, leading to the conjecture that these are the only possibilities. In this work, we conduct a systematic graph-theoretical search for integrable models within the MTLZ class. By first identifying minimal structures that a graph must contain to host an MTLZ model, we formulate an efficient algorithm to systematically search for candidate graphs for MTLZ models. Implementing this algorithm using computational software, we enumerate all candidate graphs with up to
N=13 vertices and perform an in-depth analysis of those with
N≤11. Our results corroborate the aforementioned conjecture for graphs up to
11 vertices. For even larger graphs, we propose a specific family, termed descendants of ``
(0,2)-graphs'', as promising candidates that may violate the conjecture above. Our work can serve as a guideline to identify new exactly solvable multistate Landau-Zener models in the future.