Heating and many-body resonances in a periodically driven two-band system

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.

Figure 1

Floquet realization of the interacting SSH model: the nondriven system represents a two-band model, coupled resonantly by a strong periodic drive. As a result, the ground state of the infinite-frequency Floquet Hamiltonian features an interesting topological phase.

Figure 2

Short- and long-time stroboscopic dynamics of the entropy density (a) and (c), and the energy density (b) and (d). The linewidths in (a) and (b) show the size of temporal fluctuations. The parameters are $U/J_0=1, \zeta =0.6, \delta \zeta =0.12$, and $L=20$, which in the high-frequency limit gives $J^{\prime }/J_0=0.41$ and $J/J_0=0.29$.

Figure 3

Dependence of the crossover regime on the interaction strength: excess heat (a) and excess entanglement entropy density (b). Unity on the vertical axis corresponds to an infinite-temperature state, while zero to no heating. The parameters are $\zeta =0.6, \delta \zeta =0.12$, and $L=20$, which in the high-frequency limit gives $J^{\prime }/J_0=0.41$ and $J/J_0=0.29$.

Figure 4

Energy density fluctuations as a function of the system size. The dashed green lines show the numerical data for $exp(-S_{\psi ,d}^F/2)$ at $\mathrm{\Omega }/J_0=1$ and $\mathrm{\Omega }/J_0=7$ up to $L=16$, and are extrapolated for $L>16$. Here $S_{\psi ,d}^F$ is the Floquet diagonal entropy; cf. Appendix pp1. Inset: Frequency dependence of the fluctuations at infinite time obtained using ED. The parameters are $U/J_0=1, \zeta =0.6, \delta \zeta =0.12$.

Figure 5

The energy autocorrelator $\mathcal{G}$ as a function of time (a), and its Fourier transform $\tilde{\mathcal{G}}$ (b) for $\delta \nu T=\pi /200$. The arrow shows the many-body resonances peak. (c) A pair of many-body resonant Floquet eigenstates, $|m\rangle$ and $|n\rangle$, in the crossover regime. Here $|\nu \rangle$ are the eigenstates of the approximate Floquet Hamiltonian $H_F^{\left(0\right)}$ with energy $E_{F,\nu }^{\left(0\right)}$. The vertical dashed lines mark the boundaries of the Floquet zones, while the $x$-axis range corresponds to the many-body bandwidth. A similar procedure is used in time-of-flight images of superfluid Bose gases in optical lattices where quasimomentum states are projected onto momentum states to visualize the momentum (Bragg) peaks in nearby Brillouin zones. The parameters are $U/J_0=1, \zeta =0.6, \delta \zeta =0.12$, and $L=16$.

Figure 6

Examples of nearly degenerate pairs of exact Floquet eigenstates, including Floquet many-body resonances, in the dynamical regimes of interest. The resonant state $|n\rangle$ is quantified by the quantity $R_n={\sum }_{\mu }{\left|\langle n|\mu \rangle \right|}^2\delta ({\nu }_k\le {\tilde{E}}_F^{(0+1),\mu }\le {\nu }_{k+1})$, with ${\nu }_k=k\delta \nu , k\in \mathbb{N}$, and the small energy shell $\delta \nu =\mathrm{\Omega }/100$. Here $|\mu \rangle$ denotes an eigenstate of $H_F^{(0+1)}$. The range of the $x$ axis coincides with the many-body bandwidth, while the vertical dashed lines mark the boundaries of the Floquet zones. The parameters are $U/J_0=1, \zeta =0.6, \delta \zeta =0.12$, which amounts to $J^{\prime }/J_0=0.41, J/J_0=0.29$, and $L=16$.

Figure 7

System size dependence of the exact diagonalization results. (a) Normalized energy, (b) diagonal entropy, and (c) energy density fluctuations. The parameters are $U/J_0=1, \zeta =0.6, \delta \zeta =0.12$, which amounts to $J^{\prime }/J_0=0.41, J/J_0=0.29$.

Figure 8

System size dependence of the Lanczos evolution curves. (a) Normalized energy, (b) entanglement entropy, and (c) energy density fluctuations. The parameters are $U/J_0=1, \zeta =0.6, \delta \zeta =0.12$, which amounts to $J^{\prime }/J_0=0.41, J/J_0=0.29$.

Figure 9

Comparison between infinite-time ED and long-time average of the exact Lanczos time evolution. Panels (a) and (c) show the normalized energy and energy-density fluctuations for $L=16$. In panel (a) we have assumed $E_{F,\beta =0}^{\left(0\right)}=0$. Panel (b) shows the system-size dependence of the normalized energy on a logarithmic scale. The parameters are $U/J_0=1, \zeta =0.6, \delta \zeta =0.12$, which amounts to $J^{\prime }/J_0=0.41, J/J_0=0.29$.

Figure 10

Dependence of the infinite-time normalized energy ${\overline{Q}}_{\beta }$ (a), (eigenstate) diagonal entropy ${\mathcal{S}}_{\beta }$ (b), and energy-density fluctuations ${\overline{\delta \mathcal{E}}}_{\beta }$ (c) on the temperature ${\beta }^{-1}$ of the initial state for $L=16$. The parameters are $U/J_0=1, \zeta =0.6, \delta \zeta =0.12$, which amounts to $J^{\prime }/J_0=0.41, J/J_0=0.29$.

Figure 11

Energy density ${\mathcal{E}}_{\beta }\left(200T\right)-{\mathcal{E}}_{\beta }\left(0\right)$ pumped into the system as a function of the temperature of the Gibbs initial state localized around the ground state. The parameters are $L=16, U/J_0=1, \zeta =0.6, \delta \zeta =0.12$, which amounts to $J^{\prime }/J_0=0.41, J/J_0=0.29$.

Figure 12

Frequency dependence of the normalized energy pumped into the system at infinite times ${\overline{Q}}_{\psi }$ (a), the diagonal entropy ${\mathcal{S}}_{\psi }$ (b), and the energy-density fluctuations ${\overline{\delta \mathcal{E}}}_{\psi }$ (c), starting from the ground state of the corrected Floquet Hamiltonian $H_{\mathrm{eff}}^{\left(0\right)}+H_{\mathrm{eff}}^{\left(1\right)}$, properly brought back to the laboratory frame by the leading-order kick operator $K_{\mathrm{eff}}^{\left(1\right)}\left(0\right)$. The parameters are $U/J_0=1, \zeta =0.6, \delta \zeta =0.12$, which leads to $J/J_0=0.41$ and $J^{\prime }/J_0=0.29$.

Figure 13

Frequency dependence of the mean level spacing $r_{\mathrm{ave}}=\min ({\delta }_{i+1},{\delta }_i)/\max ({\delta }_{i+1},{\delta }_i)$ in the spectra of the infinite-frequency Hamiltonian $H_{\mathrm{eff}}^{\left(0\right)}$ describing the interacting SSH model (a), the corrected Floquet Hamiltonian to leading order $H_{\mathrm{eff}}^{(0+1)}=H_{\mathrm{eff}}^{\left(0\right)}+H_{\mathrm{eff}}^{\left(1\right)}$ (b), and the exact Floquet Hamiltonian $H_{\mathrm{eff}}$ (c). The dashed horizontal $U/J_0=1, \zeta =0.6, \delta \zeta =0.12$, which amounts to $J^{\prime }/J_0=0.41, J/J_0=0.29$.