Interpretable machine learning study of the many-body localization transition in disordered quantum Ising spin chains

We apply support vector machine (SVM) to study the phase transition between many-body localized and thermal phases in a disordered quantum Ising chain in a transverse external field. The many-body eigenstate energy $E$ is bounded by a bandwidth $W=E_{\text{max}}-E_{\text{min}}$. The transition takes place on a phase diagram spanned by the energy density $\epsilon =2(E-E_{\text{min}})/W$ and the disorder strength $\delta J$ of the spin interaction uniformly distributed within $[-\delta J,\delta J]$, formally parallel to the mobility edge in Anderson localization. In our paper, we use the labeled probability density of eigenstate wave functions belonging to the deeply localized and thermal regimes at two different energy densities ($\epsilon$'s) as the training set, i.e., providing labeled data at four corners of the phase diagram. Then we employ the trained SVM to predict the whole phase diagram. The obtained phase boundary qualitatively agrees with previous work using entanglement entropy to characterize these two phases. We further analyze the decision function of the SVM to interpret its physical meaning and find that it is analogous to the inverse participation ratio in configuration space. Our findings demonstrate the ability of the SVM to capture potential quantities that may characterize the many-body localization phase transition.

Figure 1

Density of states of Hamiltonian in Eq. (1) at $\delta J=1.8$ for a specific disorder configuration. $\epsilon$ is the energy density. The mobility edges separating thermal and MBL phases are determined according to Supplemental Material of Ref. [6].

Figure 2

(a) A separating plane (solid line) separates two different phases (labeled as circles and crosses respectively) with the largest margin (shaded area) in the 2D feature space. The red circles and crosses mark the support vectors that are closest to the separating plane. (b) The large circle in the original 2D feature space is a separating hyperplane in higher dimensional space after the transformation. Such a transformation makes data points that are not linearly separable in its original space linearly separable in the transformed higher dimensional space.

Figure 3

(a) The test accuracy as a function of the order ($d$) of the polynomial kernel. The black dots denote the test accuracy. It increases from $47.5\%$ for the linear kernel at $d=1$ and approaches $100\%$ corresponding to that of the RBF kernel (the red dashed line). (b) The fraction of support vectors among all training data versus kernel order $d$ shown in the blue squares. The green dash-dot line corresponds to the fraction of SV in the RBF kernel.

Figure 4

The probability that eigen wave function corresponding to energy density $\epsilon =59/60$ generated at a given $\delta J$ is ETH phase for $\delta J\in [0,5]$. The probability is estimated using the fraction of ETH phase in an ensemble of 300 disorder realizations at energy density $\epsilon =59/60$ for $L=10$ (blue dots), $L=12$ (red dots), and $L=14$ (red dots) predicted by SVM with RBF kernel. For each size, we take the $\delta J$ corresponding to 50% probability of being ETH to be the phase boundary and denote it by $\delta J^{*}$. The inset shows the finite-size extrapolation of $\delta J^{*}$. The intercept is interpreted as the phase boundary $\delta J_c$ in the thermodynamic limit.

Figure 5

Phase diagram of the disordered quantum Ising chain defined in Eq. (1) obtained by SVM with RBF kernel. The training was performed for $0.1\le \delta J\le 0.2$ at two energy densities $\epsilon =59/60$ and $19/60$ labeled as ETH and for $8.0\le \delta J\le 10.0$ at the same two energy densities labeled as MBL. The black diamonds are the critical disorder strengths $\delta J_c$ extracted from the large $L$ extrapolations of the finite size transition points (blue, red, and green dots) at different $\epsilon$. The black dashed line is an exponential fit to the phase boundary.

Figure 6

The test accuracy obtained from 4500 testing samples of $L=12$ for the SVM machines using the linear kernel (black line) with $\mathcal{K}(\overrightarrow{x_i},\overrightarrow{x_j})=\overrightarrow{x_i}·\overrightarrow{x_j}$ and the quadratic polynomial kernel (red line) with $\mathcal{K}(\overrightarrow{x_i},\overrightarrow{x_j})={(\overrightarrow{x_i}·\overrightarrow{x_j})}^2$. The number of training samples used is indicated by the horizontal axis.

Figure 7

Distributions of $W_{ij}$ (left) and $W_{ii}$ (right) for $L=12$. When $i\ne j, W_{ij}$ can be positive or negative, but cluster very close to zero with $96.7\%$ of them distributed in the range $[-0.2,0.2]$ for an average of $-1.8*{10}^{-3}$ as denoted by the red diamond shown in the left panel. In contrast, the diagonal $W_{ii}$ are much larger. $88.6\%$ of all $W_{ii}$ are larger than 10 with an average of 22.25 as denoted by the red diamond in the right panel.

Figure 8

Fraction of data points classified as in the ETH phase in an ensemble versus the disorder strength at $\epsilon =59/60$ and $L=12$. The colored symbols and dashed lines denote results obtained by the linear SVM trained on the probability density to the $q\mathrm{th}$ power, namely $(x_1^q,x_2^q,\cdots ,x_{2^L}^q)$, where $x_i={\left|\langle {\sigma }_i^z|{\mathrm{\Psi }}_n\rangle \right|}^2$. Black dashed line is obtained using the RBF kernel trained on the original data set.

Figure 9

The ensemble-averaged half-chain entanglement entropy $S_E$ (left panel) and the participation entropy $S_P$ (right panel) plotted versus disorder strength $\delta J$ for different lengths $L$ of the quantum Ising chain at energy density $\epsilon =59/60$.

Figure 10

Schematic explanation how NN maps an input data ${\vec{x}}_i$ to its label $y_i$; the NN acts on all input data points $i=1,2,\cdots ,N$ thus plays a role as its target function.

Figure 11

The probability that eigenwave function corresponding to energy density $\epsilon =59/60$ generated at a given $\delta J$ is ETH phase for $\delta J\in [0,5]$. The probability is estimated by the fraction of ETH phase in an ensemble of 300 disorder realizations at energy density $\epsilon =59/60$ for $L=10$ (blue dots), $L=12$ (red dots), and $L=14$ (red dots) predicted by NN. For each size, we take the $\delta J$ corresponding to 50% probability of being ETH to be the phase boundary and denote it by $\delta J^{*}$. The inset shows the finite-size extrapolation of $\delta J^{*}$. The intercept is interpreted as the phase boundary $\delta J_c$ in the thermodynamic limit.

Figure 12

Phase diagram of the disordered quantum Ising chain defined in the main text. With $\epsilon =2(E-E_{\text{min}})/(E_{\text{max}}-E_{\text{min}}$) being the energy density relative to the total bandwidth. The black diamonds are $\delta J_c$ at different $\epsilon {}^{\prime }\mathrm{s}$, which are the finite size extrapolations from the finite size transition point (blue, red, and green dots).