Initial state dependent dynamics across the many-body localization transition

We investigate quench dynamics across many-body localization (MBL) transition in an interacting one-dimensional system of spinless fermions with aperiodic potential. We consider a large number of initial states characterized by the number of kinks $N_{\mathrm{kinks}}$ in the density profile, such that the equal number of sites are occupied between any two consecutive kinks. We show that on the delocalized side of the MBL transition the dynamics becomes faster with increase in $N_{\mathrm{kinks}}$ such that the decay exponent $\gamma$ in the density imbalance increases with increase in $N_{\mathrm{kinks}}$. The growth exponent of the mean square displacement which shows a power-law behavior $\langle x^2\left(t\right)\rangle \sim t^{\beta }$ in the long-time limit is much larger than the exponent $\gamma$ for one-kink and other low-kink states though $\beta \sim 2\gamma$ for a charge density wave state. As the disorder strength increases ${\gamma }_{N_{\mathrm{kink}}}\to 0$ at some critical disorder, $h_{N_{\mathrm{kinks}}}$, which is a monotonically increasing function of $N_{\mathrm{kinks}}$. A one-kink state always underestimates the value of the disorder at which the MBL transition takes place but $h_{\text{1}\text{kink}}$ coincides with the onset of the subdiffusive phase preceding the MBL phase. This is consistent with the dynamics of interface broadening for the one-kink state. We show that the bipartite entanglement entropy has a logarithmic growth $aln\left(Vt\right)$ not only in the MBL phase but also in the delocalized phase and in both the phases the coefficient $a$ increases with $N_{\mathrm{kinks}}$ as well as with the interaction strength $V$. We explain this dependence of dynamics on the number of kinks in terms of the normalized participation ratio of initial states in the eigenbasis of the interacting Hamiltonian.

Figure 1

For a half-filled system of spin-less fermions, we study various initial states characterized by the number of kinks in the density profile such that equal number of sites are occupied between any two consecutive kinks. Schematic diagram of initial states with $1,2,3$, and 5 kinks for $L=24$ sites chain is shown here.

Figure 2

The density imbalance $I\left(t\right)$ as a function of time $t$ for $h=5t_0$ and $10t_0$ at $V=t_0$ and $L=24$ for various kink initial states. The MBL transition point for the model in Eq. (1) is $h_c=6.3t_0$ at $V=t_0$ (Fig. 13). Dashed lines show the power-law fit to the form $t^{-\gamma }$. [(b) and (d)] Imbalance decay exponent $\gamma$ obtained from power-law fits for $h=5t_0$ and $10t_0$, respectively. Dashed red line in these panels is the exponent $\beta /2$ obtained from mean square displacement. Note that $\gamma \sim 0$ for all the initial states deep in the MBL phase though one can still see $\gamma$ increasing monotonically with $N_{\mathrm{kinks}}$.

Figure 3

(a) Comparison of $\gamma$ with NPR at $h=5t_0,V=t_0$ for $L=16$. Like $\gamma$, NPR also increases monotonically as we go from a one-kink state to higher kink states, being maximum for the CDW state indicating that the CDW state gets contribution from a larger fraction of eigenstates. In (b), we show $I_{\mathrm{sat}}$ along with the diagonal and off-diagonal elements of $I(t=0)$ for various kink initial states at $h=5t_0,V=t_0$ for $L=16$.

Figure 4

The density imbalance $I\left(t\right)$ as a function of time $t$ for one-, three-, seven-kink and CDW initial states for various disorder strengths at $V=t_0$ for $L=24$. (e) $\gamma$ as a function of disorder strength $h/t_0$ for various initial states. Starting from a one-kink initial state $I\left(t\right)$ shows saturation after the initial rapid decay for $h\ge h_{1\mathrm{kink}}=4t_0$ which is less than $h_c=6.3t_0$ obtained from level spacing ratio (Fig. 13), while for a higher kink state, such as a three-kink or seven-kink initial state, imbalance shows power-law decay up to much larger values of the aperiodic potential.

Figure 5

[(a) and (b)] $I_{\mathrm{diag}}$ and $I_{\text{off-diag}}(t=0)$ as a function of the disorder strength $h/t_0$ for various kink initial states for $L=16$. As the strength of aperiodic potential increases, $I_{\mathrm{diag}}$ approaches 1 first for lower kink states while the higher kink initial states require much larger $h$ to get $I_{\mathrm{diag}}\to 1$. (c) NPR of various initial states in the eigenbasis of the Hamiltonian vs $h$.

Figure 6

(a) Mean square displacement $\langle x^2\left(t\right)\rangle$ vs $t$ for various values of $V$ at $h=5t_0$. For small values of $V, \langle x^2\left(t\right)\rangle$ saturates after the initial rapid growth but for larger values of $V, \langle x^2\left(t\right)\rangle$ increases as $t^{\beta }$ in the long-time regime. (b) shows comparison of $\beta /2$ with $\gamma$ obtained from the density imbalance for various kink initial states for various values of interaction strength $V$ and $h=5t_0$. For any finite value of $V, \beta /2\gg \gamma$ for one-kink and other lower kink states. $\beta /2$ is closest to $\gamma$ obtained for larger kink states like CDW state.

Figure 7

(a) Normalized energy of the initial state $\langle E_{\mathrm{norm}}\rangle$ vs $h$ for various kink initial states at $V=t_0$ and for $L=24$. [(b)–(d)] Overlap $|c_n^2|$ of an initial state with the eigenstates of the Hamiltonian vs the normalized eigen energy $E$ for various values of $h$ for one-kink, seven-kink, and CDW initial states, respectively.

Figure 8

(a) Density imbalance $I\left(t\right)$ as a function of $t$ for a one-kink initial state while (b) shows the first moment of the particle density $m\left(t\right)$ across the interface of a one-kink for various strengths of the aperiodic potential $h$ at $V=t_0$. In (c), we show the comparison of exponents ${\gamma }_{{}_{1\mathrm{kink}}},{\gamma }_{{}_{\mathrm{CDW}}}$ and ${\gamma }^{\prime }/2$.

Figure 9

(a) shows the variance of the particle distribution and (b) presents $\mathrm{\Delta }N\left(t\right)$ vs $t$ for various values of the aperiodic potential $h/t_0$ at $V=t_0$ for a one-kink initial state. Dashed lines show the fittings of the form $t^b$, and the exponents $b_{\mathrm{\Delta }N,\mathrm{Var}}$ have been plotted as a function of disorder $h/t_0$ in the inset of (b).

Figure 10

System size analysis of $I\left(t\right)$ for various kink initial states at $h=5t_0, V=t_0$. For one-kink and three-kink initial states, the long-time decay rate of the imbalance decreases as $L$ increases resulting in either smaller values of $\gamma$ or larger saturation value of imbalance $I_{\mathrm{sat}}$. In contrast, for the CDW state, the decay gets faster in the long-time limit resulting in larger values of $\gamma$ for larger $L$. This is consistent with faster growth of $\langle x^2\left(t\right)\rangle$ in the long-time limit.

Figure 11

Disorder dependence of the bipartite entanglement entropy starting from various kink initial states. [(a)–(d)] Show $S\left(t\right)$ as a function of time $t$ for one-, three-, seven-kink and CDW initial states for various disorder strengths at $V=t_0$ for $L=24$. Dashed lines show the fitting to the logarithmic form $S\left(t\right)\sim aln\left(Vt\right)$. (e) Coefficient $a$ as a function of disorder strength $h/t_0$. Note that $h_c=6.3t_0$ from level spacing ratio (Fig. 13).

Figure 12

The bipartite entanglement entropy $S\left(t\right)-S_0\left(\infty \right)$ as a function of time $t$ starting from three-, five-, seven-kink and CDW initial states for various interaction strengths at $h=5t_0$ for $L=24$. Here $S_0\left(\infty \right)$ is the saturation value of the EE of the corresponding noninteracting system. (e) Coefficient $a$ as a function of interaction strength $V/t_0$ starting from various initial states.

Figure 13

Energy level spacing ratio of successive gaps $\langle r\rangle$ vs disorder $h$. $\langle r\rangle$ is obtained by averaging $r_n$ over the entire energy spectrum and over a large number of independent disorder configurations. Inset shows the data collapse to obatin the critical disorder $h_c\sim 6.3t_0$ with exponent $\nu \sim 1.2$. The data have been averaged ove r $10000–150$ configurations for $L=12–18$.

Figure 14

Density imbalance $I\left(t\right)$ in the localized and delocalized regime of the noninteracting system. (a) $I\left(t\right)$ as a function of $t$ starting from various kink initial states at $h=5t_0$ for the noninteracting system for $L=24$. Dynamics depends strongly on the initial states. The saturation value $I_{\mathrm{sat}}$, shown in insets, decreases as the number of kinks increases in the initial states. (b) Density imbalance for $h=t_0$ where the noninteracting system has single-particle mobility edges.

Figure 15

Density imbalance $I\left(t\right)$ in the fully delocalized regime. (a) $I\left(t\right)$ as a function of $t$ starting from various kink initial states at $h=t_0,V=t_0$ for $L=24$. The long-time fitting has been shown in (b) where power-law fits have been shown by dashed lines. In the delocalized regime, there is no monotonic increase in $\gamma$ with the number of kinks in the initial states.

Figure 16

NPR of various initial states as a function of number of kinks in the initial state at $V=t_0$ for $L=16$ sites chain. For $h=t_0$ and $h=2.1t_0$, NPR does not have a systematic dependence on the number of kinks $N_{\mathrm{kinks}}$ in initial states. This should be compared with a systematic dependence of NPR with $N_{\mathrm{kinks}}$ for $h=5t_0$.