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

TL;DR

This paper derives an exact analytical solution for the long-time dynamics of a periodically driven transverse field Ising chain, revealing how observables evolve and saturate, and extends the analysis to various driving protocols.

Contribution

It provides the first exact analytical expressions for wave functions and expectation values in a periodically driven integrable quantum system, including generalizations to different driving protocols.

Findings

Expectation values split into oscillatory and saturation parts.

Oscillatory contributions vanish at large drive cycles.

Long-time behavior described by periodic generalized Gibbs ensemble.

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.