Insights from the exact analytical solution of periodically driven transverse field Ising chain

Abstract

We derive an exact analytical expression at stroboscopic intervals for the time-dependent wave function of a class of integrable quantum many-body systems, driven by the periodic delta-kick protocol. To investigate long-time dynamics, we use the wave function to obtain an exact analytical expression for the expectation values of the defect density, magnetization, residual energy, fidelity, and the correlation function after the $n$th drive cycle. Periodically driven integrable closed quantum systems absorb energy, and the long-time universal dynamics are described by the periodic generalized Gibbs ensemble (GGE). We demonstrate that the expectation values of all observables are divided into two parts: one highly oscillatory term that depends on the drive cycle n, and the rest of the terms are independent of it. Typically, the $n$-independent part constitutes the saturation at large n and periodic GGE. The contribution from the highly oscillatory term vanishes in large $n$. We also generalize our formalism to include square pulse and sinusoidal driving protocols.