Probing many-body localization with neural networks

We show that a simple artificial neural network trained on entanglement spectra of individual states of a many-body quantum system can be used to determine the transition between a many-body localized and a thermalizing regime. Specifically, we study the Heisenberg spin-1/2 chain in a random external field. We employ a multilayer perceptron with a single hidden layer, which is trained on labeled entanglement spectra pertaining to the fully localized and fully thermal regimes. We then apply this network to classify spectra belonging to states in the transition region. For training, we use a cost function that contains, in addition to the usual error and regularization parts, a term that favors a confident classification of the transition region states. The resulting phase diagram is in good agreement with the one obtained by more conventional methods and can be computed for small systems. In particular, the neural network outperforms conventional methods in classifying individual eigenstates pertaining to a single disorder realization. It allows us to map out the structure of these eigenstates across the transition with spatial resolution. Furthermore, we analyze the network operation using the dreaming technique to show that the neural network correctly learns by itself the power-law structure of the entanglement spectra in the many-body localized regime.

Figure 1

Phase diagram of the Heisenberg chain with Hamiltonian (5) obtained from the neural network ansatz in Eq. (7) trained with cost function (8) on entanglement spectra obtained from an exact diagonalization on $N=16$ sites. The plot shows the average confidence for the MBL phase over 40 realizations of disorder as a function of the absolute maximal values of the random magnetic field $\overline{h}$, spaced with $\mathrm{\Delta }\overline{h}=0.125$, and for eigenstates belonging to different rescaled energies $\epsilon =(E-E_{min})/(E_{max}-E_{min})$. Compared to Ref. [18] where a similar plot was obtained with better-controlled, yet more sophisticated methods, we have used smaller systems and fewer disorder realizations.

Figure 2

Schematic setup of the neural network used to map entanglement spectra to the confidence with which they are classified as either belonging to the ETH or MBL regime. This map, which is explicitly given by Eq. (7), can be interpreted as the action of a hidden layer of neurons on the input data, followed by a output layer of two neurons which correspond to the two options of classification. Note that our choice of a $\mathrm{Softmax}$ activation function for the output layer implies that the confidences for ETH and MBL sum up to 1.

Figure 3

(a) Dependence of the critical value ${\overline{h}}_{\mathrm{c}}$ on how confident the network is required to be in classifying a given entanglement spectrum as MBL for the average of 40 disorder realizations of the $N=16$ chain. Here, different lines denote different percentages of MBL spectra that are required in order to classify a given $\overline{h}$ as MBL. For the transition value ${\overline{h}}_{\mathrm{c}}$, we then take the smallest $\overline{h}$ that is classified as MBL in this way. The plateaus come from the finite $\overline{h}$ resolution $\mathrm{\Delta }\overline{h}=0.125$. (b) Correlation of the critical values ${\overline{h}}_{\mathrm{c}}$ obtained from individual disorder realizations with respective mean disorder strength $\langle \left|h\right|\rangle$, averaged over all sites of the $N=16$ chain, for 40 individual disorder realizations. The correlation coefficient in this case is $\rho =\mathrm{cov}({\overline{h}}_{\mathrm{c}},\langle \left|h\right|\rangle )/[\sigma \left({\overline{h}}_{\mathrm{c}}\right)\sigma (\langle |h|\rangle )]\approx 0.76$, with $\mathrm{cov}$ denoting the covariance, and $\sigma$ the standard deviation, respectively.

Figure 4

Comparison of energy-resolved single-sample ETH to MBL transition indicators. (a) Standard deviation of the von Neumann entanglement entropy of a cut of length $N_A=9$ of the $N=18$ chain over 512 consecutive eigenstates. (b) Absolute value of the $\overline{h}$-derivative of the Schmidt gap for the same system averaged over 512 consecutive eigenstates. (c) Uncertainty in the classification of entanglement spectra by a neural network with one hidden layer, including cross-validation over 50 trainings. States classified as MBL with a confidence larger than 0.1 but smaller than 0.9 are assigned a 1, all others a 0. The continuous color range comes from averaging over 512 consecutive eigenstates. (d) Fine structure comparison of the classification of 100 representative eigenstates taken from the middle of the spectrum of a single sample of the $N=16$ chain (where $N_A=8$). The upper panel shows the value of Schmidt gap, while the lower panel shows the confidence of the neural network for classifying a given state as MBL. Note that this fine-structure analysis cannot be performed with the entanglement entropy standard deviation criterion, as this would require a coarse-graining of eigenstates.

Figure 5

Comparison of ETH to MBL transition indicators for a single realization of the disorder of the $N=18$ chain, averaged over energy density. The arrows indicate the vertical axis each curve refers to. The reduced density matrix is built for $N_A=9$. Here, the ratio $r$ of adjacent energy gaps is colored in black. For the GOE describing the ETH regime, we have $r\simeq 0.530$. For the Poisson level statistics characterizing the MBL regime, we have $r\simeq 0.386$. We compare $r$ to the energy-average of the confidence with which entanglement spectra belonging to a given value of $\overline{h}$ are classified as ETH, obtained from two neural network (NN) instances: one with the confidence-enhancing term we added to the network's cost function, corresponding to $\delta =1$ in Eq. (8), the other without it, corresponding to $\delta =0$, colored in blue and red, respectively. The large deviation in the classification of the network trained with $\delta =0$ from the established criterion of energy level statistics underlines the importance of confidence optimization. Note also that the transition is sharper for the neural network based approach with $\delta =1$.

Figure 6

Dependence of the classification of eigenstates for a single disorder realization of the $N=18$ chain on the location of the entanglement cut. (a) Close-up of the $\overline{h}$-dependent classification of 200 eigenstates taken from the middle of the spectrum for entanglement cuts ranging from $N_A=6$ up to $N_A=12$. For each value of $\overline{h}$, a subplot is shown whose horizontal axis corresponds to these seven cuts. Blue (red) states were classified with confidence $>90\%$ as ETH (MBL). The remaining states are marked in white. (b) Survey of $10000$ eigenstates from the middle of the spectrum which have been classified separately for all seven cuts. The blue (red) curve is the fraction of eigenstates that were classified with confidence $>90\%$ as ETH (MBL) for all cuts. The black curve is the fraction of states that are classified with confidence $>90\%$ as ETH for at least one cut and with the same confidence as MBL for at least one cut, i.e., states, which are spatially inhomogeneous in their character.

Figure 7

Dreaming of entanglement spectra for a single disorder realization of the $N=18$ chain (with $N_A=9$). An entry on the horizontal axis denotes the index of the respective entanglement spectrum eigenvalue. (a) 140 typical entanglement spectra of the pure ETH (blue), MBL (red), and transition (black) regime. The trained network assigns a confidence vector $(1,0)$ to the ETH spectra, and $(0,1)$ to the MBL spectra. The transition regime entanglement spectra have been chosen such that the network yields an output lying between $(0.4,0.6)$ and $(0.6,0.4)$ on them. (b) Result after $200000$ steps of the dreaming gradient descent, described in the main text, when applied on 20 of the same transition region entanglement spectra. Note the difference in scale, which indicates that the neural network learns relative rather than absolute features of the training data. The network correctly picks up the power-law structure of the entanglement spectrum in the MBL phase, as can be deduced from its nearly linear slope in this logarithmic plot.

Figure 8

Comparison of transition regions obtained from neural networks that were trained with different cost functions, corresponding to a different choice of the parameters $\delta$ and $\mu$ in Eq. (8), for a single sample of the $N=18$ chain with $N_A=9$. Here we show the classification uncertainty as defined below Fig. 4, i.e., including cross-validation over 50 trainings, for different cost functions. (a) The standard classification result with both regularization and confidence-enhancing terms in the cost function. (b) Without confidence optimization, the transition region is detected at qualitatively larger values of $\overline{h}$. From this observation alone we cannot decide which of the two networks corresponding to $\delta \in \{0,1\}$ should be preferred, since both distinguish the pure ETH and MBL regions with $\eta =1$. However, Fig. 5 clearly shows that only the cost function with $\delta =1$ faithfully recovers the transition obtained from a level statistics analysis. In addition, as discussed in Appendix pp1, when training with different kinds of input data the transition region stays unchanged only when $\delta =1$, while for $\delta =0$ the network classification becomes inconsistent. (c) and (d) Without proper regularization, the network performance becomes pathological.

Figure 9

Dependence of the classification of states on different training runs. For each plot, the entanglement spectra for a system with $N=18$ sites with a single disorder realization were trained with 50 times, each time with randomly initialized network parameters and a random choice of training data. The resulting classifications of the individual eigenstates for each $\overline{h}$ are then compared pairwise and the comparison is averaged over all pairings of trainings. We consider three categories of states: (i) classified with confidence $>90\%$ as MBL, (ii) classified with confidence $>90\%$ as ETH, (iii) others, not confidently classified. (a) Fraction of states that switch between any of the three categories (i)–(iii) to any other between two trainings. (b) Fraction of states that switch from category (i) to category (ii) (i.e., from confidently MBL to confidently ETH) or vice versa between two trainings. Note that the value of $\overline{h}$ at which these fractions assume their maximal value provides one method to determine the critical transition location ${\overline{h}}_{\mathrm{c}}$ for this particular disorder realization.

Figure 10

Finite-size dependence of the transition region for different single disorder realizations of the $N=12,14,16,18$ chains (note that we have on purpose used a different disorder realization for $N=18$ than the one used in Fig. 4 to show that the overall shape of the transition region is sample independent). Each plot shows the transition region as defined below Fig. 4 for a network trained on entanglement spectra of a cut with $N_A=N/2$. Note that the $\overline{h}$ step size for each plot differs, and is given by $\mathrm{\Delta }\overline{h}=0.01,0.05,0.125,0.125$, respectively. For each $N$, we have averaged the result over an increasing number of eigenstates in order to arrive at a resolution suited for the human eye in a controllable way. Only for $N=16$ and 18, the data are averaged over the same number of eigenstates, in order to demonstrate that the apparent improvement of the sharpness of the transition is not merely a result of a larger number of states that have been averaged over. Note also that each of (a)–(d) necessarily represent different disorder realizations and hence the value of $\overline{h}$ at which the transition occurs should not be compared between them.