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arXiv:2409.08830 · Physical review. B./Physical review. B

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

Pritam Das, Anirban Dutta

Published Jan 27, 2025• 3 citations
arXivHTMLPDF

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 $n\mathrm{th}$ 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.

DOI
10.1103/physrevb.111.045159

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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 nthn\mathrm{th}nth 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 nnn, and the rest of the terms are independent of it. Typically, the nnn-independent part constitutes the saturation at large nnn and periodic GGE. The contribution from the highly oscillatory term vanishes in large nnn. We also generalize our formalism to include square-pulse and sinusoidal driving protocols.
We show that the correlation functions of a class of periodically driven integrable closed quantum systems approach their steady-state value as ${n}^{\ensuremath{-}(\ensuremath{\alpha}+1)/\ensuremath{\beta}}$, where nnn is the number of drive cycles and $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ denote positive integers. We find that, generically, $\ensuremath{\beta}=2$ within a dynamical phase characterized by a fixed $\ensuremath{\alpha}$; however, its value can change to $\ensuremath{\beta}=3$ or $\ensuremath{\beta}=4$ either at critical drive frequencies separating two dynamical phases or at special points within a phase. We show that such decays are realized in both driven Su-Schrieffer-Heeger (SSH) and one-dimensional (1D) transverse field Ising models, discuss the role of symmetries of the Floquet spectrum in determining $\ensuremath{\beta}$, and chart out the values of $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ realized in these models. We analyze the SSH model for a continuous drive protocol using a Floquet perturbation theory which provides analytical insight into the behavior of the correlation functions in terms of its Floquet Hamiltonian. This is supplemented by an exact numerical study of a similar behavior for the 1D Ising model driven by a square pulse protocol. For both models, we find a crossover timescale nc{n}_{c}nc​ which diverges at the transition. We also unravel a long-time oscillatory behavior of the correlators when the critical drive frequency, ${\ensuremath{\omega}}_{c}$, is approached from below $(\ensuremath{\omega}<{\ensuremath{\omega}}_{c})$. We tie such behavior to the presence of multiple stationary points in the Floquet spectrum of these models and provide an analytic expression for the time period of these oscillations.
We study the long time dynamics in closed quantum systems periodically driven via time dependent parameters with two frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}=r{\ensuremath{\omega}}_{1}$. Tuning of the ratio rrr there can unleash plenty of dynamical phenomena to occur. Our study includes integrable models like Ising and XYXYXY models in d=1d=1d=1 and the Kitaev model in d=1d=1d=1 and 2 and can also be extended to Dirac fermions in graphene. We witness the wave-function overlap or dynamic freezing that occurs within some small/ intermediate frequency regimes in the $({\ensuremath{\omega}}_{1},r)$ plane (with $r\ensuremath{\ne}0$) when the ground state is evolved through a single cycle of driving. However, evolved states soon become steady with long driving, and the freezing scenario gets rarer. We extend the formalism of adiabatic-impulse approximation for many cycle driving within our two-rate protocol and show the near-exact comparisons at small frequencies. An extension of the rotating wave approximation is also developed to gather an analytical framework of the dynamics at high frequencies. Finally we compute the entanglement entropy in the stroboscopically evolved states within the gapped phases of the system and observe how it gets tuned with the ratio rrr in our protocol. The minimally entangled states are found to fall within the regime of dynamical freezing. In general, the results indicate that the entanglement entropy in our driven short-ranged integrable systems follow a genuine nonarea law of scaling and show a convergence (with a rrr dependent pace) towards volume scaling behavior as the driving is continued for a long time.
We study an integrable Hamiltonian reducible to free fermions, which is subjected to an imperfect periodic driving with the amplitude of driving (or kicking), randomly chosen from a binary distribution like a coin-toss problem. The randomness present in the driving protocol destabilizes the periodic steady state reached in the limit of perfectly periodic driving, leading to a monotonic rise of the stroboscopic residual energy with the number of periods (N)(N)(N) for such Hamiltonians. We establish that a minimal deviation from the perfectly periodic driving in the present case using such protocols would always result in a bounded heating up of the system with NNN to an asymptotic finite value. Exploiting the completely uncorrelated nature of the randomness and the knowledge of the stroboscopic Floquet operator in the perfectly periodic situation, we provide an exact analytical formalism to derive the disorder averaged expectation value of the residual energy through a disorder operator. This formalism not only leads to an immense numerical simplification, but also enables us to derive an exact analytical form for the residual energy in the asymptotic limit which is universal, i.e., independent of the bias of coin-toss and the protocol chosen. Furthermore, this formalism clearly establishes the nature of the monotonic growth of the residual energy at intermediate NNN while clearly revealing the possible nonuniversal behavior of the same.
We study the work statistics of a periodically-driven integrable closed quantum system, addressing in particular the role played by the presence of a quantum critical point. Taking the example of a one-dimensional transverse Ising model in the presence of a spatially homogeneous but periodically time-varying transverse field of frequency , we arrive at the characteristic cumulant generating function G ( u ), which is then used to calculate the work distribution function P ( W ). By applying the Floquet theory we show that, in the infinite time limit, P ( W ) converges, starting from the initial ground state, towards an asymptotic steady state value whose small- W behaviour depends only on the properties of the small-wave-vector modes and on a few important ingredients: the time-averaged value of the transverse field, h 0 , the initial transverse field, , and the equilibrium quantum critical point , which we find to generate a sequence of non-equilibrium critical points , with l integer. When , we find a ‘universal’ edge singularity in P ( W ) at a threshold value of which is entirely determined by . The form of that singularity—Dirac delta derivative or square root—depends on h 0 being or not at a non-equilibrium critical point h * l . On the contrary, when , G ( u ) decays as a power-law for large u , leading to different types of edge singularity at . Generalizing our calculations to the case in which we initialize the system in a finite temperature density matrix, the irreversible entropy generated by the periodic driving is also shown to reach a steady state value in the infinite time limit.
Does a closed quantum many-body system that is continually driven with a time-dependent Hamiltonian finally reach a steady state? This question has only recently been answered for driving protocols that are periodic in time, where the long-time behavior of the local properties synchronizes with the drive and can be described by an appropriate periodic ensemble. Here, we explore the consequences of breaking the time-periodic structure of the drive with additional aperiodic noise in a class of integrable systems. We show that the resulting unitary dynamics leads to new emergent steady states in at least two cases. While any typical realization of random noise causes eventual heating to an infinite-temperature ensemble for all local properties in spite of the system being integrable, noise that is self-similar in time leads to an entirely different steady state (which we dub the ``geometric generalized Gibbs ensemble'') that emerges only after an astronomically large time scale. To understand the approach to the steady state, we study the temporal behavior of certain coarse-grained quantities in momentum space that fully determine the reduced density matrix for a subsystem with size much smaller than the total system. Such quantities provide a concise description for any drive protocol in integrable systems that are reducible to a free-fermion representation.
Abstract In this work we study the entanglement entropy of a uniform quantum Ising chain in transverse field undergoing a periodic driving of period τ . By means of Floquet theory we show that, for any subchain, the entanglement entropy tends asymptotically to a value τ -periodic in time. We provide a semi-analytical formula for the leading term of this asymptotic regime: It is constant in time and obeys a volume law. The entropy in the asymptotic regime is always smaller than the thermal one: because of integrability the system locally relaxes to a generalized Gibbs ensemble (GGE) density matrix. The leading term of the asymptotic entanglement entropy is completely determined by this GGE density matrix. Remarkably, the asymptotic entropy shows marked features in correspondence to some non-equilibrium quantum phase transitions undergone by a Floquet state analog of the ground state.