Quench dynamics in a model with tuneable integrability breaking

We consider quantum quenches in an integrable quantum chain with tuneable-integrability-breaking interactions. In the case where these interactions are weak, we demonstrate that at intermediate times after the quench local observables relax to a prethermalized regime, which can be described by a density matrix that can be viewed as a deformation of a generalized Gibbs ensemble. We present explicit expressions for the approximately conserved charges characterizing this ensemble. We do not find evidence for a crossover from the prethermalized to a thermalized regime on the time scales accessible to us. Increasing the integrability-breaking interactions leads to a behavior that is compatible with eventual thermalization.

Figure 1

Green's function $G_0(j,l,t)$ for a quench with ${\delta }_i=0.75$, ${\delta }_f=0.25$, and a lattice with $L=100$ sites.

Figure 2

Comparison between the mode occupation numbers $\langle n_{+}\left(k\right)\rangle$ for Gibbs and generalized Gibbs ensembles for a quench with ${\delta }_i=0.75$, ${\delta }_f=0.25$. The effective inverse temperature for this quench is ${\beta }_{\mathrm{eff}}=2.95782J$.

Figure 3

Comparison between the mode occupation numbers $\langle n_{-}\left(k\right)\rangle$ for Gibbs and generalized Gibbs ensembles for a quench with ${\delta }_i=0.75$, ${\delta }_f=0.25$. The effective inverse temperature for this quench is ${\beta }_{\mathrm{eff}}=2.95782J$.

Figure 4

Green's function $\langle c_{L/2}^{†}{c}_{L/2+j}\rangle$ calculated in the Gibbs and generalized Gibbs ensembles for a quench with ${\delta }_i=0.75$, ${\delta }_f=0.25$, and a lattice with $L=100$ sites. The effective inverse temperature for this quench is ${\beta }_{\mathrm{eff}}=2.95782J$.

Figure 5

A schematic of the CUT method for finding the approximate time evolution of the operator O to order $U$.

Figure 6

Comparison of exact (solid) $U=0$ nearest-neighbour Green's function $\mathcal{G}(L/2,L/2+1)=\langle c_{L/2}{c}_{L/2+1}^{†}\rangle$ with the CUT results for the quench ${\delta }_i=0.8\to \delta =0.4$ and $U_i=0\to U$ on the $L=100$ chain.

Figure 7

Comparison of exact (solid) $U=0$ next-nearest-neighbor Green's function $\mathcal{G}(L/2,L/2+2)=\langle c_{L/2}{c}_{L/2+2}^{†}\rangle$ with the CUT results for the quench ${\delta }_i=0.8\to \delta =0.4$ and $U_i=0\to U$ on the $L=100$ chain.

Figure 8

A comparison of the CUT Green's function $\left|\mathcal{G}\right(100,101,t)-\mathcal{G}(100,101,t\to \infty \left)\right|$ and the free fermion asymptotic form, Eq. (19), on the $L=200$ chain for the quench ${\delta }_i=0.75\to \delta =0.5$ and $U_i=0\to U=0.15$. The prefactor of the power law $t^{-3/2}$ is used as a fit parameter. The revival time of the $L=200$ chain is $t\sim 50$ and the asymptotic value $\mathcal{G}(100,101,t\to \infty )=-0.482275$.

Figure 9

A comparison between the free fermion asymptotic form of the Green's function, Eq. (19), and the CUT result for the quench ${\delta }_i=0.75\to \delta =0.5$ and $U_i=0\to U=0.15$ on the $L=200$ chain. The prefactor of the power law $t^{-3/2}$ is used as a fit parameter.

Figure 10

Comparison of the CUT and t-DMRG results for $\mathcal{G}(L/2,L/2+1)=\langle c_{L/2}{c}_{L/2+1}^{†}\rangle$ for the quench ${\delta }_i=0.75\to \delta =0.5$ and $U_i=0\to$ $U=0.15$ on a $L=50$ chain. The revival time for the $L=50$ system is ${\tau }_r\sim 13$.

Figure 11

Comparison of the CUT and t-DMRG results for $\mathcal{G}(L/2,L/2+1)=\langle c_{L/2}{c}_{L/2+1}^{†}\rangle$ for the quench ${\delta }_i=0.75\to \delta =0.5$ and $U_i=0\to$ $U=0.25$ on a $L=50$ chain.

Figure 12

Comparison of the CUT and t-DMRG results for $\mathcal{G}(L/2,L/2+1)=\langle c_{L/2}{c}_{L/2+1}^{†}\rangle$ for the quench ${\delta }_i=0.75\to \delta =0.5$ and $U_i=0\to$ $U=0.5$ on a $L=50$ chain.

Figure 13

Rescaled difference between the t-DMRG and CUT data for $\mathcal{G}(25,26)$ and different values of $U$.

Figure 14

$\mathcal{G}(L/2,L/2+2)$ for the quench ${\delta }_i=0.75\to \delta =0.5$, $U_i=0\to U=0.15$ on an $L=50$ chain.

Figure 15

$\mathcal{G}(L/2,L/2+3)$ after the quench ${\delta }_i=0.75\to \delta =0.5$, $U_i=0\to U=0.15$ on an $L=50$ chain.

Figure 16

Comparison of the CUT and t-DMRG results for $\mathcal{G}(L/2,L/2+4)$ for the quench ${\delta }_i=0.75\to \delta =0.5$, $U_i=0\to U=0.15$ on an $L=50$ chain.

Figure 17

The $L$ dependence of the difference between the deformed GGE and the $t\to \infty$ CUT result for the four-point function for a number of separations. The solid lines are linear fits $c{L}^{-1}$ to the data.

Figure 18

Nearest neighbor density-density correlation function $\langle n\left(\frac{L}{2}\right)n(\frac{L}{2}+1)\rangle$ for a quench from ${\delta }_i=0.8\to {\delta }_f=0.4$ and $U=0\to 0.4$ computed by t-DMRG for system size $L=100$. For comparison we show CUT results for $L=40$ and the asymptotic value predicted by the $L=50$ deformed GGE.

Figure 19

Next-nearest-neighbor density-density correlation function $\langle n\left(\frac{L}{2}\right)n(\frac{L}{2}+2)\rangle$ for a quench from ${\delta }_i=0.8\to {\delta }_f=0.4$ and $U=0\to 0.4$ computed by t-DMRG for system size $L=100$. The correlator relaxes to a stationary value consistent with the deformed GGE prediction (evaluated for $L=50$).

Figure 20

Time evolution of $\mathcal{G}(L/2,L/2+1)$ for quenches with ${\delta }_i=0.8\to {\delta }_f=0.4$ and $U_i=0\to U=0.4$ and system sizes $L=16$, $L=100$, and $L=200$ sites. The data for $L=16$ are ED results for systems with periodic boundary conditions (PBCs) and are seen to exhibit many revivals.

Figure 21

Time evolution of $\mathcal{G}(L/2,L/2+1)$ for quenches with ${\delta }_i=0.8\to {\delta }_f=0.4$ and $U_i=0\to U=0.4$. $L=50$ site system up to $t\sim 200$, with error bars estimated in Appendix pp2.

Figure 22

Comparison of the t-DMRG, time-averaged (t-av) ED, and QMC results for the nearest-neighbor Green's function at time $t$ after the quench ${\delta }_i=0.8\to {\delta }_f=0.4$, $U=0\to 0.4$. t-DMRG and QMC simulations are performed on the $L=100$ chain, while ED studies the $L=16$ chain.

Figure 23

Comparison of the t-DMRG, time-averaged (t-av) ED, and QMC results for the nearest-neighbor Green's function at time $t$ after the quench ${\delta }_i=0.8\to {\delta }_f=0.4$, $U=0\to 1$. t-DMRG and QMC simulations are performed on the $L=100$ chain, while ED studies the $L=16$ chain.

Figure 24

Comparison of the t-DMRG, time-averaged (t-av) ED, and QMC results for the nearest-neighbor Green's function at time $t$ after the quench ${\delta }_i=0.8\to {\delta }_f=0.4$, $U=0\to 2$. t-DMRG simulations have been performed on the $L=100,200$ chain, QMC results are presented for the $L=100$ chain, and ED studies the $L=16$ chain.

Figure 25

Comparison of the t-DMRG, time-averaged (t-av) ED, and QMC results for the nearest-neighbor Green's function at time $t$ after the quench ${\delta }_i=0.8\to {\delta }_f=0.4$, $U=0\to 3$. t-DMRG and QMC simulations are performed on the $L=100$ chain, while ED studies the $L=16$ chain.

Figure 26

Comparison of the t-DMRG, time-averaged (t-av) ED, and QMC results for the nearest-neighbor Green's function at time $t$ after the quench ${\delta }_i=0.8\to {\delta }_f=0.4$, $U=0\to 4$. t-DMRG and QMC simulations are performed on the $L=100$ chain, while ED studies the $L=16$ chain.

Figure 27

Comparison of the t-DMRG, time-averaged (t-av) ED, and QMC results for the nearest-neighbor Green's function at time $t$ after the quench ${\delta }_i=0.8\to {\delta }_f=0.4$, $U=0\to 6$. t-DMRG and QMC simulations are performed on the $L=100$ chain, while ED studies the $L=16$ chain.

Figure 28

Comparison of the t-DMRG, time-averaged (t-av) ED, and QMC results for the nearest-neighbor Green's function at time $t$ after the quench ${\delta }_i=0.8\to {\delta }_f=0.4$, $U=0\to 8$. t-DMRG and QMC simulations are performed on the $L=100$ chain, while ED studies the $L=16$ chain.

Figure 29

Comparison of the t-DMRG, time-averaged (t-av) ED, and QMC results for the nearest-neighbor Green's function at time $t$ after the quench ${\delta }_i=0.8\to {\delta }_f=0.4$, $U=0\to 10$. t-DMRG simulations have been performed on the $L=100,200$ chain, QMC results are presented for the $L=100$ chain, and ED studies the $L=16$ chain.

Figure 30

Difference in the value of $\mathcal{G}(L/2,L/2+1)$ between finite-temperature results obtained with QMC ($L=100$) or ED ($L=16$) to the time-average values obtained via ED for $L=16$ for a quench with ${\delta }_i=0.8\to {\delta }_f=0.4$ and $U_i=0\to U_f$ as a function of $U_f$. Finite-size effects are less pronounced for small values of $U_f$, but prominent for $U_f>1$. The intermediate region $1\le U_f<8$ is the best candidate to obtain thermalization on long time scales in this system.

Figure 31

Comparison of t-DMRG results for the time evolution of $\mathcal{G}(L/2,L/2+1)$ for systems with $L=100$ sites for quenches with initial $U_i=0,{\delta }_i=0.8$ to values of $U_f=10$ and ${\delta }_f=0.4$ or ${\delta }_f=0$, respectively. As can be seen, the expectation value for both cases is very similar.

Figure 32

Green's function results from t-DMRG and ED for the quench ${\delta }_i=0.8\to {\delta }_f=0.4$ with $U_i=5$ to $U=0,0.2$. As with Figs. 22, 23, 24, 25, 26, 27, 28, 29 we see that the time-averaged (t-av) ED is compatible (up to finite-size effects) with the t-DMRG plateau value, while the thermal expectation is not.

Figure 33

Green's function results from t-DMRG and ED for the quench ${\delta }_i=0.8\to {\delta }_f=0.4$ with $U_i=5$ to $U=0.5,1.0$. As with Figs. 22, 23, 24, 25, 26, 27, 28, 29 we see that the time-averaged (t-av) ED is compatible (up to finite-size effects) with the t-DMRG plateau value.

Figure 34

Differences between runs with different parameters and different quenches ($L=100$ in all cases).

Figure 35

Error estimates for t-DMRG results on the time evolution of $\mathcal{G}(L/2,L/2+1)$ for a system with $L=200$ sites and a quench ${\delta }_i=0.75\to {\delta }_f=0.5$ and $U_i=0\to U_f=0.5$. The data are obtained using a time step of $\delta t=0.005$ and a target discarded weight of $\varepsilon ={10}^{-9}$. The red error bars (lines) are obtained from the estimate discussed in Appendix pp2; the blue ones (asterisks) are obtained by comparing to the results of a run with time step $\delta t=0.01$. The error estimate appears to be larger, but of similar order of magnitude to the actual deviation between the results at times $\sim$50. From this estimate we obtain at the end of the time evolution a relative error of the order of 1.5%.