Entanglement dynamics of a many-body localized system coupled to a bath

The combination of strong disorder and interactions in closed quantum systems can lead to many-body localization (MBL). However, this quantum phase is not stable when the system is coupled to a thermal environment. We investigate how MBL is destroyed in systems that are weakly coupled to a dephasive Markovian environment by focusing on their entanglement dynamics. We numerically study the third Rényi negativity $R_3$, a recently proposed entanglement proxy based on the negativity that captures the unbounded logarithmic growth in the closed case and that can be computed efficiently with tensor networks. We also show that the decay of $R_3$ follows a stretched exponential law, similarly to the imbalance, with, however, a smaller stretching exponent.

Figure 1

A sketch of our setup: a spin chain with disordered $z$-directed fields $h_i\in [-W,W]$, spin exchange $\mathrm{\Delta }$, and coupling $\mathrm{\Gamma }$ to a bath.

Figure 2

A sketch of the $R_3$ entanglement probe, see Eq. (12), that we compute using tensor-network techniques. We make a bipartition of the system into two subsystems $A$ and $B$, and partially transpose the degrees of freedom of subsystem $B$ before taking the trace.

Figure 3

Quench dynamics of $R_3=2S_3$ in the closed quantum system at fixed system size $L=14$ for various interaction strengths. In the inset we show the finite-size scaling of $R_3$ at fixed interaction strength $\mathrm{\Delta }=1$, from bottom to top $L=10,12,14,16$. The error bars show the standard error of the mean.

Figure 4

(a) Comparison of the average of $R_3$ obtained by the numerics over 2500 disorder realizations with system size $L=16$ and $\mathrm{\Delta }=1$ and by the model described in the text with ${\xi }_1=0.40$ and ${\xi }_2=0.195$. The dashed grey line shows the curve of the model shifted by a constant. (b)–(d) Normalized histograms of the model and numerics at several times. The dotted vertical lines indicate the mean, while the dashed green line indicates the resonance at $2log2$. The tails in the distributions of the model are more pronounced, as it does not take into account multispin couplings.

Figure 5

Dynamics of the Rényi negativity (top) and the imbalance (bottom) under a quench in the open or closed system ($L=20$). The disorder is fixed to $W=5J$ and the interaction strength is $\mathrm{\Delta }=1$ for the MBL system, and $\mathrm{\Delta }=0$ for the noninteracting Anderson localized system. The two darkest colors (red and blue) show the closed system for comparison. At intermediate timescales and for sufficiently weak dephasing strength, $R_3$ distinguishes MBL from single-particle localization. A feature that is absent in the imbalance. In the insets we show the interquartile range (IQR) which forms a measure for the spread of the distribution.

Figure 6

Histograms at two different times (columns) corresponding to the averages shown in Fig. 5. (a), (b) Interacting system with $\mathrm{\Delta }=1$. (c), (d) Noninteracting system $\mathrm{\Delta }=0$. The green line indicates $R_3=2log2$ which corresponds to the maximal entanglement between 2-spins.

Figure 7

Quench dynamics of the Rényi negativity $R_3$ in a system with disorder $W=5J$ and coupling $\mathrm{\Gamma }=0.1J$. Top: MBL with interaction strength $\mathrm{\Delta }=1$. Bottom: Noninteracting system $\mathrm{\Delta }=0$. In the inset we show the best fitting parameter for the stretching exponent $b$ with $3\sigma$ errorbars obtained by the least-square method. These exponents do not depend on the interactions up to leading order. The blue line shows one of the fitting functions with $b\approx 0.25$.

Figure 8

Relative error of the fourth-order TEBD scheme for various simulation parameters for a small system of $L=8$ spins. In the left column the coupling is $\mathrm{\Gamma }=0.01J$, in the right one $\mathrm{\Gamma }=1J$. In the top line we vary the maximal bond dimension. In the bottom line we vary the time step (in units of $J^{-1}$) at maximal bond dimension $\chi =256$.

Figure 9

Relative error of the fourth-order TEBD scheme for $L=40$ spins with a coupling strength of $\mathrm{\Gamma }=0.01J$. On the left we show the relative difference between simulations with bond dimension $\chi$ and bond dimension ${\chi }_{max}=500$ with a time step of $dt=0.05$. On the right we show the difference between simulation with time step is $dt=0.1$ and $\mathrm{d}t=0.05$ at bond dimension $\chi$.

Figure 10

Negativity dynamics in a small system of $L=8$ spins. When spin-conservation is broken by adding a term $g{\sum }_i{S}_i^x$ to the Hamiltonian, the negativity dynamics stops abruptly at a finite time. In the left plot $g=0.2J$, and in the right one $g=0.4J$.

Figure 11

The logarithmic negativity $\mathcal{E}$ and the third Rényi negativity $R_3$ for the two-qubit Werner state defined by $\lambda$. $R_3$ is not an entanglement monotone as it takes a nonzero value in the separable regime $\lambda <1/3$.

Figure 12

Comparison of the negativity (top) and the third Rényi negativity (bottom) in the Lindblad quench dynamics in the $XXZ$model at disorder $W=5J$, interaction strength $\mathrm{\Delta }=1$ and small system size $L=8$. The dashed blue line shows the stretched exponential fits $e^{-{\left(\frac{\mathrm{\Gamma }t}{a}\right)}^b}$ for the decay. The stretching exponents are equal for both quantities. The insets show the same data in a different scale. The error bars show the standard error of the mean and the averages are taken over 1000 disorder realizations.

Figure 13

The quench dynamics of the Fisher information density $f_Q=F_Q/L$ at disorder $W=5J$, interaction strength $\mathrm{\Delta }=1$, and system size $L=8$. We show stretched exponential fits $e^{-{\left(\frac{\mathrm{\Gamma }t}{a}\right)}^b}$ for the decay that start approximately at time $\sim \frac{1}{\mathrm{\Gamma }}$. The inset shows the same data in a different scale. The error bars show the standard error of the mean and the averages are taken over 1000 disorder realizations.