Many-body localization and delocalization in large quantum chains

We theoretically study the quench dynamics for an isolated Heisenberg spin chain with a random on-site magnetic field, which is one of the paradigmatic models of a many-body localization transition. We use the time-dependent variational principle as applied to matrix product states, which allows us to controllably study chains of a length up to $L=100$ spins, i.e., much larger than $L\simeq 20$ that can be treated via exact diagonalization. For the analysis of the data, three complementary approaches are used: (i) determination of the exponent $\beta$ which characterizes the power-law decay of the antiferromagnetic imbalance with time; (ii) similar determination of the exponent ${\beta }_{\mathrm{\Lambda }}$ which characterizes the decay of a Schmidt gap in the entanglement spectrum; and (iii) machine learning with the use, as an input, of the time dependence of the spin densities in the whole chain. We find that the consideration of the larger system sizes substantially increases the estimate for the critical disorder $W_c$ that separates the ergodic and many-body localized regimes, compared to the values of $W_c$ in the literature. On the ergodic side of the transition, there is a broad interval of the strength of disorder with slow subdiffusive transport. In this regime, the exponents $\beta$ and ${\beta }_{\mathrm{\Lambda }}$ increase, with increasing $L$, for relatively small $L$ but saturate for $L\simeq 50$, indicating that these slow power laws survive in the thermodynamic limit. From a technical perspective, we develop an adaptation of the “learning by confusion” machine-learning approach that can determine $W_c$.

Figure 1

(a) Spin imbalance (3) as a function of time $t$ for a spin chain of length $L=100$ and various disorder strengths $W=2,4,8$. The blue line shows the disorder average over $R=\mathcal{O}\left({10}^3\right)$ realizations and the dotted black line shows the noninteracting case $\mathrm{\Delta }=0$ using 10 000 independent disorder realizations. The red (green) line shows the most (least) localized single realization. (b) Probability distribution function of the fluctuations of the imbalance around the average $\mathcal{J}\left(t\right)\equiv \mathcal{I}\left(t\right)-\overline{\mathcal{I}\left(t\right)}$ for $W=2$ (top), $W=8$ (bottom), $L=50$ (green), and $L=100$ (blue) over the time interval $t\in [50,100]$. The solid (dashed) red line shows a Gaussian fit for $L=100$ ($L=50$).

Figure 2

Power-law exponent $\beta$ corresponding to the decay of the imbalance $\mathcal{I}\propto t^{-\beta }$ over the window $t\in [50,100]$, for various system sizes $L$. For the cases $L=50$ and 100, the ten sites near each edge of the chain were not considered. The square symbol shows the result for an independent implementation of the time-evolving block decimation (TEBD) algorithm [48]. Triangles indicate a weak finite-time decay of ${\mathcal{I}}_{\mathrm{A}}$ for the noninteracting case $\mathrm{\Delta }=0$, $L=100$. Error bars are $1\sigma$ intervals based on a bootstrapping procedure.

Figure 3

(a) The von Neumann entropy of entanglement $S$ as a function of time for the same parameters as in Fig. 1. (b) Distributions of $S$ over the interval $t\in [95,100]$. The blue (green) lines show the result for $L=100$ ($L=50$).

Figure 4

(a) The Schmidt gap $\mathrm{\Lambda }$ as a function of time for the same parameters as in Fig. 1. (b) Distributions of $\mathrm{\Lambda }$ over the interval $t\in [95,100]$. The blue (green) lines show the result for $L=100$ ($L=50$).

Figure 5

Power-law exponent ${\beta }_{\mathrm{\Lambda }}$ corresponding to the decay of the disorder-averaged Schmidt gap $\mathrm{\Lambda }\propto t^{-{\beta }_{\mathrm{\Lambda }}}$ over the window $t\in [50,100]$, for various system sizes $L$. Error bars are $1\sigma$ intervals based on a bootstrapping procedure.

Figure 6

Results of the supervised machine-learning algorithm. Shown is the average confidence $C$ with which time series (spin densities $\langle S^z\rangle$ as a function of time) taken from TDVP for a given disorder magnitude are classified as belonging to the localized phase, where we labeled $W=8$ as localized and $W=2$ as delocalized. A plateau emerges at large $W$ for $L=100$, indicating a transition to the MBL regime. The rightmost panel shows $1-C$ on a semilogarithmic scale. Error bars indicate the standard deviation of the distribution of $C$ and are $1\sigma$ intervals based on 100 independent training sessions (these should be understood as being cut off at $C=0$ and $C=1$).

Figure 7

Results of the unsupervised machine-learning algorithm, based on “learning by confusion” for input data with $L=16$, $L=50$, and $L=100$. Shown is the distribution of the accuracy $\langle A\left(W_0\right)\rangle$ over 100 training instances with different initial conditions for each $W_0$. For system sizes $L=50$ and $L=100$, the distribution is bimodal in the interval $3.25<W_0<5.5$ with the two maxima tracing out the red lines as a function of $W_0$.

Figure 8

Time evolution of the disorder-averaged imbalance (3) as computed by the TDVP with a truncated bond dimension $\chi =32,64$ and by exact numerics $\chi =256$ for a small system $L=16$, up to $t=300$. We consider several hundred independent realizations of disorder. For the sake of comparison, also the noninteracting case $\mathrm{\Delta }=0$ is shown.

Figure 9

Time evolution of the disorder-averaged imbalance (3) as computed by the TDVP with bond dimension $\chi =32,64,96$ for a moderately large system $L=50$ up to $t=200$ and choices of disorder strength $W=2,3,5$. For comparison, also the noninteracting case $\mathrm{\Delta }=0$ is shown. The fitted coefficients for the power-law decay of the imbalance in the window $t\in [100,200]$ for $W=5$ are $\beta (\chi =32)=0.025\pm 0.005$, $\beta (\chi =64)=0.020\pm 0.005$, $\beta (\chi =96)=0.020\pm 0.004$, which is in good agreement (within error bars) with the value found for $t\in [50,100]$ as presented in the main text. Insets show a zoomed region in the time window $t\in [50,200]$ on a log-log scale.

Figure 10

Top: time evolution of the disorder-averaged imbalance (3) in the case of $L=100$, $W=2$ (left), and $W=5$ (right), for various choices of the bond dimension $\chi$. We consider several hundred independent realizations of disorder. For the sake of comparison, also the noninteracting case $\mathrm{\Delta }=0$ is shown, computed both using the TDVP as well as using exact diagonalization. Bottom: time evolution of the disorder-averaged entropy for the same parameter choices. The horizontal dashed lines indicate the cutoff of the entropy $S_{\mathrm{cutoff}}={log}_2\chi$. The shaded region indicates the error ($2\sigma$ intervals) in the average entropy computed from the standard deviation of the data for the case $\chi =64$.

Figure 11

Power-law exponent $\beta$ corresponding to the decay of the imbalance $\mathcal{I}\propto t^{-\beta }$ over the window $t\in [50,100]$, for $L=100$. Both the average and typical values are considered. Triangles indicate a weak finite-time decay of ${\mathcal{I}}_{\mathrm{A}}$ for the noninteracting case $\mathrm{\Delta }=0$, $L=100$. Error bars are $2\sigma$ intervals based on a bootstrapping procedure. The vertical gray line indicates the putative transition from the ergodic to many-body localized regime based on the vanishing of $\beta$ within $2\sigma$, subtracting the systematic error.