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arXiv:2409.17053 · SciPost Physics

Knizhnik-Zamolodchikov equations and integrable hyperbolic Landau-Zener models

Suvendu Barik, Lieuwe Bakker, Vladimir Gritsev, Emil A. Yuzbashyan

Published Jun 27, 2025• 2 citations
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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 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.

DOI
10.21468/scipostphys.18.6.212

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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 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.
Abstract We consider a set of non-stationary quantum models. We show that their dynamics can be studied using links to Knizhnik-Zamolodchikov (KZ) equations for correlation functions in conformal field theories. We specifically consider the boundary Wess-Zumino-Novikov-Witten model, where equations for correlators of primary fields are defined by an extension of KZ equations and explore the links to dynamical systems. As an example of the workability of the proposed method, we provide an exact solution to a dynamical system that is a specific multi-level generalization of the two-level Landau-Zenner system known in the literature as the Demkov-Osherov model. The method can be used to study the nonequilibrium dynamics in various multi-level systems from the solution of the corresponding KZ equations.
In this paper we consider the problem of solving quantum field theories with time dependent interaction strengths. We show that the recently formulated framework [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)], which is a generalization of the regular Bethe ansatz technique, provides the exact many-body wavefunction. In this framework, the time-dependent Schrodinger equation is reduced to a set of analytic difference equations and matrix difference equations, called the quantum Knizhnik-Zamolodchikov (qKZ) equations. The consistency of the solution gives rise to constraints on the time-dependent interaction strengths. For interaction strengths satisfying these constraints, the system is integrable, and the solution to the qKZ and the analytic difference equations provides the explicit form of the many-body wavefunction that satisfies the time-dependent Schrodinger equation. We provide a concrete example by considering the SU(2)SU(2)SU(2) Gross-Neveu model with time dependent interaction strength. Using this framework we solve the model with the most general time-dependent interaction strength and obtain the explicit form of the wave function.
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.
Quantum integrability has been applied to a large variety of low dimensional Hamiltonians in Quantum Field Theory, Condensed Matter Physics, and Statistical Mechanics to obtain exact expressions for the spectrum and thermodynamics of these systems. In most of these studies the coupling constants are constant in time. Here we present an exact solution of the nonstationary Schrödinger equation for the Kondo Hamiltonian with a time-dependent spin-exchange coupling J(t) J ( t ) of the form \lambda t + p(t) ± \sqrt{(\lambda t + p(t))^2 + 16/3} λ t + p ( t ) ± ( λ t + p ( t ) ) 2 + 16 / 3 , where p(t) p ( t ) is an arbitrary periodic function, under periodic boundary concditions. Unlike previously studied time-dependent integrable models, which are rooted in the classical Yang–Baxter structure and associated Knizhnik–Zamolodchikov equations, our approach is based on the quantum Knizhnik–Zamolodchikov framework and the quantum Yang–Baxter algebra. Our results broaden the domain of time-dependent integrability to a genuinely quantum class of models and provide new tools for exploring coherent nonequilibrium dynamics in strongly correlated systems.