Heating Rates in Periodically Driven Strongly Interacting Quantum Many-Body Systems

We study heating rates in strongly interacting quantum lattice systems in the thermodynamic limit. Using a numerical linked cluster expansion, we calculate the energy as a function of the driving time and find a robust exponential regime. The heating rates are shown to be in excellent agreement with Fermi’s golden rule. We discuss the relationship between heating rates and, within the eigenstate thermalization hypothesis, the smooth function that characterizes the off-diagonal matrix elements of the drive operator in the eigenbasis of the static Hamiltonian. We show that such a function, in nonintegrable and (remarkably) integrable Hamiltonians, can be probed experimentally by studying heating rates as functions of the drive frequency.

Figure 1

(Main panels) Absolute value of the energy per site $|e(\tau )|$ vs $\tau$ for (a) the nonintegrable and (b) the integrable ${\widehat{H}}_0$ for three strengths $g=\{0.05,0.2,0.8\}$ of the drive, a period $T=1.0$, and ${\beta }_I=(30{)}^{-1}$. Results (at stroboscopic times) are obtained using NLCE to (a) 16 (NLCE-16) and 17 (NLCE-17) orders, and (b) 17 (NLCE-17) and 18 (NLCE-18) orders. The solid lines show exponential fits to the highest NLCE order. (Insets) Rates obtained in fits, as those depicted in the main panels, for the two highest NLCE orders. For all values of $g$, the fits for the nonintegrable ${\widehat{H}}_0$ are done for times $3\le \tau \le 20$ for NLCE-17 and $3\le \tau \le 15$ for NLCE-16, while for the integrable ${\widehat{H}}_0$ they are done for times $2\le \tau \le 8$ for NLCE-18 and $2\le \tau \le 7$ for NLCE-17. The Fermi golden rule predictions (open symbols) are evaluated using full exact diagonalization in chains with: (a) 17 and 18 sites (Fermi-17 and Fermi-18) and (b) 19 and 20 sites (Fermi-19 and Fermi-20), and periodic boundary conditions. Error bars indicate the fitting errors for the NLCE rates, and the standard deviation from averages over different values of $\mathrm{\Delta }E$ and $\tau$ for the Fermi golden rule predictions [54]. Power-law fits ($\alpha g^{\gamma }$) of the rates in both insets are done for the highest order of the NLCE in the interval $0.05\le g\le 0.3$.

Figure 2

(Main panel) Absolute value of the energy per site $|e(\tau )|$, normalized by its initial value $|e(0)|$, for a periodically driven nonintegrable ${\widehat{H}}_0$ with $g=0.5$ and $T=1.0$, for initial thermal states of ${\widehat{H}}_I$ at different inverse temperatures ${\beta }_I$. We show results for 16 and 17 orders of the NLCE (NLCE-16 and NLCE-17, respectively), and exponential fits to the NLCE-17 results. (Inset) Rates obtained from exponential fits to NLCE-17 for $3\le \tau \le 20$ (as those in the main panel) and NLCE-16 for $3\le \tau \le 15$ vs ${\beta }_I$, for $g=0.2$ and $g=0.5$. We also report Fermi’s golden rule predictions obtained using full exact diagonalization in chains with 17 and 18 sites (Fermi-17 and Fermi-18) and periodic boundary conditions.

Figure 3

(Main panels) Heating rates (normalized by $g^2$) vs $\mathrm{\Omega }$ for (a) the nonintegrable and (b) the integrable ${\widehat{H}}_0$, for $g=0.2$ and $g=0.3$. Rates obtained from exponential fits of the dynamics (as in the insets) are shown as symbols for NLCE to (a) 15 (NLCE-15) and 16 (NLCE-16) orders, and (b) to 17 (NLCE-17) and 18 (NLCE-18) orders. Rates obtained from Eq. (5) evaluated using full exact diagonalization in periodic chains are shown as lines for (a) 18 (${\mathrm{\Gamma }}^{18}$) and 19 (${\mathrm{\Gamma }}^{19}$) sites, and (b) 20 (${\mathrm{\Gamma }}^{20}$) and 21 (${\mathrm{\Gamma }}^{21}$) sites. We also show rates of the Fourier mode $m=1$ in Eq. (5) for (a) 19 [${\mathrm{\Gamma }}_{m=1}^{19}$] and (b) 21 [${\mathrm{\Gamma }}_{m=1}^{21}$] sites, as well as exponential fits of the results at high $\mathrm{\Omega }$. (Insets) Absolute value of the energy per site $|e(\tau )|$ vs $\tau$, using NLCE to (a) 15 (NLCE-15) and 16 (NLCE-16) orders and (b) 17 (NLCE-17) and 18 (NLCE-18) orders, for $g=0.3$ and three different driving periods $T=2\pi /\mathrm{\Omega }$. Exponential fits to the highest order of the NLCE are shown as solid lines. The rates reported in the main panels are obtained from exponential fits for (a) $3\le \tau \le 15$ for NLCE-16 and $3\le \tau \le 12$ for NLCE-15, and (b) $2\le \tau \le 8$ for NLCE-18 and $2\le \tau \le 7.5$ for NLCE-17, for all $g$ and $T$ (error bars indicate fitting errors).