Abstract The concept of quantum integrability has been introduced recently for quantum systems with explicitly time-dependent Hamiltonians (Sinitsyn et al 2018 Phys. Rev. Lett . 120 190402). Within the multistate Landau–Zener (MLZ) theory, however, there has been a successful alternative approach to identify and solve complex time-dependent models (Sinitsyn and Chernyak 2017 J. Phys. A: Math. Theor . 50 255203). Here we compare both methods by applying them to a new class of exactly solvable MLZ models. This class contains systems with an arbitrary number of interacting states and shows quick growth with N number of exact adiabatic energy crossing points, which appear at different moments of time. At each N , transition probabilities in these systems can be found analytically and exactly but complexity and variety of solutions in this class also grow with N quickly. We illustrate how common features of solvable MLZ systems appear from quantum integrability and develop an approach to further classification of solvable MLZ problems.
The WKB analysis for higher order differential equations had not been developed until recent progress of the so-called exact WKB method, so it was difficult to find WKB solutions for multistate nonadiabatic models in general, whereas enormous efforts were made especially for the study of generalized Landau-Zener models. Under such circumstances Aoki-Kawai-Takei provided a closed expression for the
S matrix of a time-dependent multistate nonadiabatic system, in which diabatic potential curves are given as polynomial functions of time and they all intersect with each other. Their result is completely explicit, and the model covers much wider classes of systems than multistate nonadiabatic models studied so far. Here we numerically show how their exact WKB formula works as long as crossing points of diabatic potential curves are well separated. The model studied includes the cases where diabatic potential functions are nonlinear, not like generalized Landau-Zener models. We also discuss the situations where virtual turning points and new Stokes curves, new components in the Stokes geometry appearing only for higher order differential equations, come into play. In particular, we examine how the presence of new Stokes curves affects final transition probabilities using a couple of concrete examples.
Abstract All currently known exactly solvable multistate Landau–Zener (MLZ) models are associated with families of operators that commute with the MLZ Hamiltonians and depend on time linearly. There can also be operators that satisfy the integrability conditions with the MLZ Hamiltonians but depend on time quadratically. We show that, among the MLZ systems, such time-quadratic operators are much more common. We demonstrate then that such operators generally lead to constraints on the independent variables that parametrize the scattering matrix. Such constraints lead to asymptotically exact expressions for the transition probabilities in the adiabatic limit of a three-level MLZ model. New more complex fully solvable MLZ systems are also found.
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.
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.