Quantum Integrability of Hamiltonians with Time-Dependent Interaction Strengths and the Renormalization Group Flow

TL;DR

This work demonstrates that integrability constraints for quantum Hamiltonians with time-dependent interaction strengths exactly mirror the renormalization-group (RG) flow of the corresponding static Hamiltonians, by mapping time t to the RG scale via t=log Λ. In the anisotropic Kondo model, the authors derive explicit flow equations for the couplings that coincide with known RG equations in the appropriate limits, and show that integrable time dependences are those along RG trajectories. They construct the exact time-dependent solution using a generalized Bethe ansatz, derive the associated qKZ equations, and reveal a universal relation between integrability and RG flow that persists across regularization schemes, including the long-time limit J(t)→π/t. The results suggest a generic principle: for any integrable Hamiltonian with fixed couplings, time-dependent driving preserves integrability if and only if couplings follow their RG equations, providing a bridge between dynamical control and RG structure with potential broad implications for driven quantum systems.

Abstract

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 $J_{\parallel}(t)$ and $J_{\perp}(t)$. 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 $J_{\parallel}(t)$ and $J_{\perp}(t)$, 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.