Long-time Behavior of Isolated Periodically Driven Interacting Lattice Systems

We study the dynamics of isolated interacting spin chains that are periodically driven by sudden quenches. Using full exact diagonalization of finite chains, we show that these systems exhibit three distinct regimes. For short driving periods, the Floquet Hamiltonian is well approximated by the time-averaged Hamiltonian, while for long periods, the evolution operator exhibits properties of random matrices of a circular ensemble (CE). In between, there is a crossover regime. Based on a finite-size scaling analysis and analytic arguments, we argue that, for thermodynamically large systems and nonvanishing driving periods, the evolution operator always exhibits properties of the CE of random matrices. Consequently, the Floquet Hamiltonian is a nonlocal Hamiltonian with multispin interaction terms, and the driving leads to the equivalent of an infinite temperature state at long times. These results are connected to the breakdown of the Magnus expansion and are expected to hold beyond the specific lattice model considered.

Popular Summary

Physical systems that are driven periodically in time, for example, by mechanical shaking or incident light, can exhibit surprising properties that do not exist when the systems are not driven. An example is a Kapitza pendulum, which is a classical rigid pendulum with a vertically oscillating point of suspension. When the shaking is fast enough, the pendulum becomes stable in the upward position (with the pendulum balanced on its support), which, in the absence of driving, would be unstable. Another spectacular example is the superconducting behavior of some materials at room temperature, which can be induced by shining high-power lasers on an otherwise normal metal. All of these surprising properties are encoded in the Floquet Hamiltonian, which is an effective, time-independent description valid for any periodically driven system.

We investigate the properties of the Floquet Hamiltonian for interacting driven quantum lattice systems and show that these properties change qualitatively as a function of the driving period. We recover three distinct regimes as a function of driving. For short driving periods, the Floquet Hamiltonian is well approximated by the time-averaged Hamiltonian; for long periods, the evolution operator exhibits the properties of the random matrices of the circular orthogonal ensemble. In this case, the system absorbs energy from the driving and heats up indefinitely. The third regime is characterized by a crossover between the time-averaged Hamiltonian and a circular orthogonal ensemble. Moreover, the Floquet Hamiltonian exhibits many-body nonlocal interactions, which means that it is qualitatively different from typical static Hamiltonians that are local and only involve few-body interactions. Even though we consider only a specific lattice model, we expect that our results should also be applicable to other models.

Our work highlights that periodically driven quantum systems can be drastically different from unitarily evolving but undriven systems.

Figure 1

Probability distribution $P(r)$ for the ratio of two consecutive energy gaps in the spectrum of ${\widehat{H}}_{\text{ave}}$ [see Eq. (12)] together with the GOE and POI predictions. (Inset) The error distance between $P(r)$ and the GOE prediction decreases quickly with an increasing system size.

Figure 2

Average value of $r$ vs $T$. The values of $⟨r⟩$ (symbols) and $⟨r_{\text{ave}}⟩$ (dashed lines) are compared to the COE, GOE, and POI predictions. The COE and GOE predictions are expected to coincide in the thermodynamic limit (see Appendix pp2). Vertical dotted lines depict the periods $T_1=2\pi \hslash /W$, $T_2=\pi \hslash /\sigma$, and $T_3=2\pi \hslash /\sigma$ (see text).

Figure 3

Distribution $P(r)$ for the periods $T=T_1$, $T_2$, and $T_3$ shown in Fig. 2. We compare the exact phases obtained from the exact diagonalization of ${\widehat{U}}_{\text{cycle}}$ (indicated by red circles) and the phases one obtains if ${\widehat{H}}_F={\widehat{H}}_{\text{ave}}$ (indicated by squares), with the GOE, COE, and POI predictions (lines).

Figure 4

Average entropy of the eigenstates $⟨S⟩$ vs the period $T$. $⟨S⟩/\ln (0.48D)$ crosses over from zero to the COE prediction, which is reached earlier as $L$ increases. The variance $⟨(S_n-⟨S⟩{)}^2{⟩}^{1/2}$ is smaller than the size of the symbols used.

Figure 5

Absorbed energy. (Main) Absorbed energy at long (infinite in our calculation) times $Q$ [see Eq. (18)] vs the driving period $T$. The initial state is a thermal state (with respect to ${\widehat{H}}_{\text{ave}}$) with $\beta =0.5$. (Inset) Expectation values $⟨{\varphi }_n|{\widehat{H}}_{\text{ave}}|{\varphi }_n⟩$ vs the exact phases ${\theta }_n$ for $L=24$ and $T=6$.

Figure 6

Expectation values $⟨{\varphi }_n|{\widehat{H}}_{\text{ave}}|{\varphi }_n⟩$ vs the exact phases ${\theta }_n$ folded in $[-\pi ,\pi )$ for $L=24$ and different driving periods $T$.

Figure 7

Transformation between ${\theta }_1$, ${\theta }_2$, ${\theta }_3$ and $x$, ${\theta }_2$, $z$.

Figure 8

The distribution $P(r)$ in the Gaussian and circular ensembles of random matrix theory and for a sequence of Poisson distributed random numbers. The label POI refers to Poisson, while the labels OE, UE, and SE refer to orthogonal, unitary, and symplectic ensembles, respectively. Continuous lines depict results for the Gaussian ensembles, while dotted lines depict results for the circular ensembles.