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arXiv:1708.07324

Breaking the integrability of the Heisenberg model through periodic driving

Pieter W. Claeys, Jean-Sébastien Caux

Published Aug 24, 2017• 6 citations
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Abstract

We study the fate of interacting quantum systems which are periodically driven by switching back and forth between two integrable Hamiltonians. This provides an unconventional and tunable way of breaking integrability, in the sense that the stroboscopic time evolution will generally be described by a Floquet Hamiltonian which progressively becomes less integrable as the driving frequency is reduced. Here, we exemplify this idea in spin chains subjected to periodic switching between two integrable anisotropic Heisenberg Hamiltonians. We distinguish the integrability-breaking effects of resonant interactions and perturbative (local) interactions, and illustrate these by contrasting different measures of energy in Floquet states and through a study of level spacing statistics. This scenario is argued to be representative for general driven interacting integrable systems.

DOI
10.48550/arxiv.1708.07324

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We study the fate of interacting quantum systems which are periodically driven by switching back and forth between two integrable Hamiltonians. This provides an unconventional and tunable way of breaking integrability, in the sense that the stroboscopic time evolution will generally be described by a Floquet Hamiltonian which progressively becomes less integrable as the driving frequency is reduced. Here, we exemplify this idea in spin chains subjected to periodic switching between two integrable anisotropic Heisenberg Hamiltonians. We distinguish the integrability-breaking effects of resonant interactions and perturbative (local) interactions, and illustrate these by contrasting different measures of energy in Floquet states and through a study of level spacing statistics. This scenario is argued to be representative for general driven interacting integrable systems.
When a closed quantum system is driven periodically with period T, it approaches a periodic state synchronized with the drive in which any local observable measured stroboscopically approaches a steady value. For integrable systems, the resulting behavior is captured by a periodic version of a generalized Gibbs ensemble. By contrast, here we show that for generic nonintegrable interacting systems, local observables become independent of the initial state entirely. Essentially, this happens because Floquet eigenstates of the driven system at quasienergy ω(α) consist of a mixture of the exponentially many eigenstates of the undriven Hamiltonian, which are thus drawn from the entire extensive undriven spectrum. This is a form of equilibration which depends only on the Hilbert space of the undriven system and not on any details of its Hamiltonian.
Adiabatically varying the driving frequency of a periodically driven many-body quantum system can induce controlled transitions between resonant eigenstates of the time-averaged Hamiltonian, corresponding to adiabatic transitions in the Floquet spectrum and presenting a general tool in quantum many-body control. Using the central spin model as an application, we show how such controlled driving processes can lead to a polarization-based decoupling of the central spin from its decoherence-inducing environment at resonance. While it is generally impossible to obtain the exact Floquet Hamiltonian in driven interacting systems, we exploit the integrability of the central spin model to show how techniques from quantum quenches can be used to explicitly construct the Floquet Hamiltonian in a restricted many-body basis and model Floquet resonances.
We consider a quantum system periodically driven with a strength which varies slowly on the scale of the driving period. The analysis is based on a general formulation of the Floquet theory relying on the extended Hilbert space. It is shown that the dynamics of the system can be described in terms of a slowly varying effective Floquet Hamiltonian that captures the long-term evolution, as well as rapidly oscillating micromotion operators. We obtain a systematic high-frequency expansion of all these operators. Generalizing the previous studies, the expanded effective Hamiltonian is now time dependent and contains extra terms appearing due to changes in the periodic driving. The same applies to the micromotion operators which exhibit a slow temporal dependence in addition to the rapid oscillations. As an illustration, we consider a quantum-mechanical spin in an oscillating magnetic field with a slowly changing direction. The effective evolution of the spin is then associated with non-Abelian geometric phases reflecting the geometry of the extended Floquet space. The developed formalism is general and also applies to other periodically driven systems, such as shaken optical lattices with a time-dependent shaking strength, a situation relevant to the cold-atom experiments.
We give a general overview of the high-frequency regime in periodically driven systems and identify three distinct classes of driving protocols in which the infinite-frequency Floquet Hamiltonian is not equal to the time-averaged Hamiltonian. These classes cover systems, such as the Kapitza pendulum, the Harper–Hofstadter model of neutral atoms in a magnetic field, the Haldane Floquet Chern insulator and others. In all setups considered, we discuss both the infinite-frequency limit and the leading finite-frequency corrections to the Floquet Hamiltonian. We provide a short overview of Floquet theory focusing on the gauge structure associated with the choice of stroboscopic frame and the differences between stroboscopic and non-stroboscopic dynamics. In the latter case, one has to work with dressed operators representing observables and a dressed density matrix. We also comment on the application of Floquet Theory to systems described by static Hamiltonians with well-separated energy scales and, in particular, discuss parallels between the inverse-frequency expansion and the Schrieffer–Wolff transformation extending the latter to driven systems.
We study the dynamics and stability in a strongly interacting resonantly driven two-band model. Using exact numerical simulations, we find a stable regime at large driving frequencies where the time evolution is governed by a local Floquet Hamiltonian that is approximately conserved out to very long times. For slow driving, on the other hand, the system becomes unstable and heats up to infinite temperature. While thermalization is relatively fast in these two regimes (but to different ``temperatures''), in the crossover between them we find slow nonthermalizing time evolution: temporal fluctuations become strong and temporal correlations long lived. Microscopically, we trace back the origin of this nonthermalizing time evolution to the properties of rare Floquet many-body resonances, whose proliferation at lower driving frequency removes the approximate energy conservation, and thus produces thermalization to infinite temperature.
An integrable model subjected to a periodic driving gives rise generally to a nonintegrable Floquet Hamiltonian. Here we show that the Floquet Hamiltonian of the integrable Lieb-Liniger model in the presence of a linear potential with a periodic time-dependent strength is instead integrable and its quasienergies can be determined using the Bethe ansatz approach. We discuss various aspects of the dynamics of the system at stroboscopic times and we also propose a possible experimental realization of the periodically driven tilting in terms of a shaken rotated ring potential.