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arXiv:2608.31080

Boundary Quantum Knizhnik-Zamolodchikov Equations and Integrability of Quantum Field Theories with Time-Dependent Bulk and Boundary Coupling Strengths

Parameshwar R. Pasnoori

math-phcond-mat.str-elhep-thquant-phSubmitted Aug 31, 2026• Announced Sep 1, 2026• 0 citations
v1 · Aug 31, 2026v2 · Sep 17, 2026
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Abstract

The generalized Bethe ansatz framework formulated in [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)] provides a unified framework to find exact solutions to quantum many-body systems with time-dependent coupling strengths with periodic boundary conditions. In this work we extend this framework to the case of open boundary conditions where in addition to the time-dependent interactions in the bulk, the boundary conditions are explicitly time-dependent. We show that for integrable time-dependent bulk coupling strengths, the generalized Bethe ansatz framework provides the time-dependent boundary conditions compatible with integrability and reduces the time-dependent Schrodinger equation to a set of matrix difference equations called the boundary quantum Knizhnik-Zamolodchikov (BqKZ) equations. The solution to the BqKZ equations provides the explicit form of the exact wavefunction. We further show that the RG invariants of the corresponding static model identify with the dynamical invariants in the time-dependent model.

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The generalized Bethe ansatz framework formulated in [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)] provides a unified framework to find exact solutions to quantum many-body systems with time-dependent coupling strengths with periodic boundary conditions. In this work we extend this framework to the case of open boundary conditions where in addition to the time-dependent interactions in the bulk, the boundary conditions are explicitly time-dependent. We show that for integrable time-dependent bulk coupling strengths, the generalized Bethe ansatz framework provides the time-dependent boundary conditions compatible with integrability and reduces the time-dependent Schrodinger equation to a set of matrix difference equations called the boundary quantum Knizhnik-Zamolodchikov (BqKZ) equations. The solution to the BqKZ equations provides the explicit form of the exact wavefunction. We further show that the RG invariants of the corresponding static model identify with the dynamical invariants in the time-dependent model.
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.
In this paper we consider quantum Hamiltonians with time-dependent interaction strengths, and following the recently formulated generalized Bethe ansatz framework [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)], we show that constraints imposed by integrability take the same form as the renormalization group flow equations corresponding to the respective Hamiltonians with constant interaction strengths. As a concrete example, we consider the anisotropic time-dependent Kondo model characterized by the time-dependent interaction strengths J∥(t)J_{\parallel}(t)J∥​(t) and J⊥(t)J_{\perp}(t)J⊥​(t). We construct an exact solution to the time-dependent Schrodinger equation and by applying appropriate boundary conditions on the fermion fields we obtain a set of matrix difference equations called the quantum Knizhnik-Zamolodchikov (qKZ) equations corresponding to the XXZ R-matrix. The consistency of these equations imposes constraints on the time-dependent interaction strengths J∥(t)J_{\parallel}(t)J∥​(t) and J⊥(t)J_{\perp}(t)J⊥​(t), such that the system is integrable. Remarkably, the resulting temporal trajectories of the couplings are shown to coincide exactly with the RG flow trajectories of the static Kondo model, establishing a direct and universal correspondence between integrability and renormalization-group flow in time-dependent quantum systems.
In this Letter we consider the time-dependent Kondo model where a magnetic impurity interacts with the electrons through a time-dependent interaction strength J(t)J(t)J(t). We develop an alternative framework based on the Bethe ansatz and construct an exact solution to the time-dependent Schr\"odinger equation. We show that when periodic boundary conditions are applied, the consistency of the solution results in a constraint equation which relates the amplitudes corresponding to a certain ordering of the particles in the configuration space. This constraint equation takes the form of a matrix difference equation, and the associated consistency conditions restrict the interaction strength J(t)J(t)J(t) for the system to be integrable. For a given J(t)J(t)J(t) satisfying these constraints, the solution to the matrix difference equations provides the exact many-body wave function that satisfies the time-dependent Schr\"odinger equation. We provide a concrete example of J(t)J(t)J(t) which satisfies these constraint equations. We show that in this case, the matrix difference equations turn into quantum Knizhnik-Zamolodchikov (qKZ) equations, which are well studied in the literature. The framework developed in this Letter allows one to probe the nonequilibrium physics of the Kondo model, and being general, it also allows one to solve a new class of Hamiltonians with time-dependent interaction strengths which are based on quantum Yang-Baxter algebra.
It is well established that in time-dependent quantum systems, integrability preserving time-dependent interaction strengths are identical to the renormalization group (RG) trajectories of the corresponding static model when time `ttt' in the driven model is identified with the logarithm of the cutoff `log⁡Λ\logΛlogΛ' of the static model. We refer to this integrability preserving driving as the RG protocol. In this work we extend the class of time-dependent integrable models to include non-Hermitian quantum models with time-dependent interaction strengths. Using the recently formulated generalized Bethe ansatz framework [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)], we show that the constraints imposed by integrability are more general: The interaction strengths of the static model that flow in the RG follow the respective RG trajectories in the corresponding time-dependent model as described above. In addition, the interaction strengths of the static model that are RG invariant can either be constant or have a specific time-dependence in the corresponding time-dependent model which is constrained by integrability. Thus we establish that in the context of time-dependent non-Hermitian systems, the set of integrability preserving time-dependent strengths is larger than the set corresponding to the RG protocol.
We present a general method to construct and solve time-dependent integrable chiral quantum field theories on finite intervals from autonomous scattering and reflection data. Spatial homogeneity forces the spectral labels of right- and left-moving particles to be affine with a common slope, so bulk and boundary couplings sample one spectral trajectory. A complete round trip shifts the spectral labels and gives boundary quantum Knizhnik--Zamolodchikov equations. The Yang--Baxter and reflection equations provide local consistency, while boundary-qKZ flatness provides global consistency. We reconstruct the local Hamiltonian by Cayley inversion of the bulk and boundary matching matrices. We apply the construction to rational SU(N)SU(N)SU(N) scattering with fundamental boundary impurities and to an O(3)O(3)O(3) model with a spin-12\tfrac1221​ impurity. For the SU(2)SU(2)SU(2) impurity, we construct explicit boundary-qKZ solutions as lattice sums of off-shell boundary Bethe vectors and recover the static open-chain Bethe equations when the drive rate vanishes
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.