Entanglement entropy of disordered quantum chains following a global quench

We numerically investigate the growth of the entanglement entropy $S_{\mathrm{ent}}\left(t\right)$ in time $t$, after a global quench from a product state, in quantum chains with various kinds of disorder. The main focus is, in particular, on fermionic chains with bond disorder. In the noninteracting case at criticality we numerically test recent predictions by the real-space renormalization group for the entanglement growth in time, the maximal entanglement as a function of block size, and the decay of a density-wave order parameter. We show that multiprecision calculations are required to reach the scaling regime and perform such calculations for specific cases. For interacting models with binary bond disorder we present data based on infinite-size density matrix renormalization group calculations and exact diagonalizations. We obtain numerical evidence of a many-body localized phase in bond-disordered systems where $S_{\mathrm{ent}}\left(t\right)\sim lnt$ seems to hold. Our results for bond disorder are contrasted with the well-studied case of potential disorder.

Figure 1

(a) $S_{\mathrm{ent}}\left(t\right)$ for the open XX chain without disorder. Symbols denote ED data for system sizes $N=50,100,200,400,800,1600,3200,6400,12800$ (from bottom to top); solid lines are a guide to the eye. The dashed vertical lines denote the crossover scale $t^{*}=\ell /2$. (b) Symbols show the saturation value as a function of block size $\ell$. The fits in (a) and (b) show that $S_{\mathrm{ent}}\sim at$ for $t<t^{*}$ and a saturation at $S_{\mathrm{ent}}\sim \frac{a}{2}\ell$ for $t>t^{*}$.

Figure 2

$S_{\mathrm{ent}}$ for the open XX chain with box potential disorder $W=0.1$. The dashed black lines denote the crossover scale $t^{*}=\ell /2$ in the clean case. Inset: Saturation value $S_{\mathrm{ent}}\left(\ell \right)$ for $t\to \infty$. The fits show that $S_{\mathrm{ent}}\sim at$ for $t<t^{*}$ and $S_{\mathrm{ent}}\sim \frac{a}{2}\ell$ for $t>t^{*}$ still approximately hold if $N<3200$.

Figure 3

$S_{\mathrm{ent}}$ for the open XX chain with box potential disorder $W=1.0$. The dashed vertical lines denote the crossover scale $t^{*}=\ell /2$ in the clean system. Inset: Saturation value versus block length $\ell =N/2$.

Figure 4

Disorder averaged stationary probability distribution ${\left|\mathrm{\Psi }(x,t\to \infty )\right|}^2$ for a single particle initially located in the middle of a chain ($x=0$) with $N=3200$ sites and disorder $W=1.0$.

Figure 5

XX model with potential disorder $W$ and $N=1000$ using 2000 samples: (a) DOS for $W=0.0,1.0$, and $W=10.0$ using a bin size $\mathrm{\Delta }\varepsilon =|{\varepsilon }_{\max }|/100$. (b) and (c) Localization measure $I_{\varepsilon }$, Eq. (9), with $m=10$.

Figure 6

Density-wave order parameter $\mathrm{\Delta }n\left(t\right)$. The circles denote the numerical data. For weak disorder $W=0.1$ the result closely resembles the clean case (solid line). For $W=1.0$ the data are well fitted by an exponential decay to a finite value, Eq. (12) (solid line).

Figure 7

Density-wave order parameter $\mathrm{\Delta }n\left(t\right)$. The decay for $W=2.0$ and $W=10.0$ is still exponential; however, the decay rate $\tau$ is a nonmonotonic function of disorder. Lines are fits of the asymptotic behavior using Eq. (12).

Figure 8

$S_{\mathrm{ent}}$ for the open XX chain with binary potential disorder $W=1.0$. The dashed vertical lines denote the crossover scale $t^{*}=\ell /2$ in the clean system. Note that the curves for $N=400$ and $N=800$ are almost on top of each other.

Figure 9

Density-wave order parameter $\mathrm{\Delta }n\left(t\right)$ for infinite binary disorder. The dashed line denotes the long-time mean $\overline{\mathrm{\Delta }n}=1/3$.

Figure 10

Open XX chain with $N=1000$ sites and bond disorder $J_i\in (0,1)$ using 1000 samples. (a) DOS based on 100 energy bins. Inset: Scaling ${\left[\right|\varepsilon \left|\rho \left(\varepsilon \right)\right]}^{-1/3}\sim |ln{\varepsilon }^2|$ [see Eq. (15)] near $\left|\varepsilon \right|\sim 0$ using 1800 bins. (b) $I_{\varepsilon }$ with $m=10$: delocalization of eigenstates, $I_{\varepsilon }\to 0$ for $\varepsilon \to 0$ in the thermodynamic limit.

Figure 11

$S_{\mathrm{ent}}\left(t\right)$ for the XX model with bond disorder $J_i\in (0,1)$ and PBC, obtained using standard double-precision arithmetics. The numbers in parentheses in the legend refer to the number of samples. The dashed line is a fit. Inset: Saturation value as a function of block length $\ell =N/2$ with circles denoting results in double precision and diamonds denoting results in double-double precision.

Figure 12

Double-precision results: Saturation value $S_{\mathrm{ent}}(\ell ,t\to \infty )$ for a quench from the initial state (3) in the periodic XX chain with $J_i\in (0,1)$ and various chain lengths $N$ and block sizes $\ell$ as indicated. The dashed line is a fit representing potential logarithmic scaling.

Figure 13

Same as Fig. 11, but with open instead of periodic boundary conditions. The $N=3200$ data are fitted, and a $ln(lnt)$ scaling is observed. Inset: Saturation value as a function of block length $\ell =N/2$.

Figure 14

Same as Fig. 11, but with quenching from a random initial product state instead of the density-wave state (3). Inset: Saturation value, with circles denoting results in double precision and diamonds denoting results in double-double precision.

Figure 15

$S_{\mathrm{ent}}\left(t\right)$ for the XX model with bond disorder $J_i\in (0,1)$, PBC, and quenching from the density-wave state. The line is a fit, showing that the initial increase is almost logarithmic in time.

Figure 16

Decay of $\mathrm{\Delta }n\left(t\right)$, Eq. (10), in the XX model with bond disorder $J_i\in [0,1), N=800$ sites, PBC, and averaging over 18 000 samples. (a) Short-time behavior and (b) long-time decay.

Figure 17

$S_{\mathrm{ent}}\left(t\right)$ for the XX model with binary bond disorder $\delta =0.4$ and OBC. Fit of the $N=1600$ data (dashed line). Inset (a) shows saturation value $S_{\mathrm{ent}}(\ell ,t\to \infty )$, and inset (b) shows approximately logarithmic scaling of $S_{\mathrm{ent}}\left(t\right)$ for $t\le 1000$.

Figure 18

$\mathrm{\Delta }n\left(t\right)$ for the XX model with $N=800$ sites, 40 000 samples, and binary bond disorder $\delta$ as indicated. The lines are a guide to the eye.

Figure 19

Critical random transverse Ising model with $J_i\in (0,1)$ and $h_i\in (0,1)$: (a) DOS as defined in Eq. (8) using 10 000 samples with 100 bins (main) and 3400 bins (inset). (b) Localization measure (9) with $m=10$.

Figure 20

$S_{\mathrm{ent}}\left(t\right)$ in the random critical transverse Ising model with PBC for a quench from the initial state (3) with block size $\ell =N/2$. Inset: Saturation value $S_{\mathrm{ent}}(\ell ,t\to \infty )$.

Figure 21

Saturation value $S_{\mathrm{ent}}(\ell ,t\to \infty )$ for a quench from the initial state (3) in the critical transverse Ising chain for various chain lengths $N$ as indicated. The dashed line is a fit representing potential logarithmic scaling.

Figure 22

(a) $\mathrm{\Delta }n\left(t\right)$ for binary disorder $\delta =0.2$ and different interaction strengths $\mathrm{\Delta }$ obtained by LCRG for an infinite chain. The dashed line for $\mathrm{\Delta }=0$ is the exact diagonalization result; the dashed line for $\mathrm{\Delta }=8$ is a fit. (b) $\mathrm{\Delta }n\left(t\right)$ for $\mathrm{\Delta }=1$ (solid line) and a fit of the envelope (dashed line). (c) Same as in (b), but for $\mathrm{\Delta }=2$.

Figure 23

LCRG data for ${\tilde{S}}_{\mathrm{ent}}\left(t\right)$ of a system consisting of real sites and ancilla sites which realize binary bond disorder with $\delta =0.92$ exactly. Subsequent curves are shifted by $0.2$.

Figure 24

Exact diagonalization data for the isotropic XXZ model ($\mathrm{\Delta }=2.0$) with binary bond disorder $\delta =0.92$. Averages are performed over 15 000 samples for $L<16$ and 11 600 samples for $L=16$.