Learning disordered topological phases by statistical recovery of symmetry

We apply the artificial neural network in a supervised manner to map out the quantum phase diagram of disordered topological superconductors in class DIII. Given the disorder that keeps the discrete symmetries of the ensemble as a whole, translational symmetry which is broken in the quasiparticle distribution individually is recovered statistically by taking an ensemble average. By using this, we classify the phases by the artificial neural network that learned the quasiparticle distribution in the clean limit and show that the result is totally consistent with the calculation by the transfer matrix method or noncommutative geometry approach. If all three phases, namely the ${\mathbb{Z}}_2$, trivial, and thermal metal phases, appear in the clean limit, the machine can classify them with high confidence over the entire phase diagram. If only the former two phases are present, we find that the machine remains confused in a certain region, leading us to conclude the detection of the unknown phase which is eventually identified as the thermal metal phase.

Figure 1

The architecture of a feedforward artificial neural network with two hidden layers, at which the input data are compressed to extract some abstract feature for classification. The activation of the output layer is the softmax function so that the sum is unity, allowing us to interpret the confidence of the machine.

Figure 2

Typical single-shot quasiparticle distribution of the first excited state, $P\left(\mathbf{r}\right)$, and its disorder average, $\langle P\left(\mathbf{r}\right)\rangle$, over 500 realizations of random configurations. The parameters are taken from deep inside the phases as $(\mu ,W)=(2,9),(6,5),(2,18)$ with $\mathrm{\Delta }=3$ and ${\mathrm{\Delta }}_2=0$ from the top. The system size is taken as $10\times 10$.

Figure 3

(a) Average outputs of 200 ternary-classifying ANNs trained with the clean-limit data for $t=1,\mathrm{\Delta }=3,{\mathrm{\Delta }}_2=2$. The color of each point $(\mu ,W)$ indicates the confidence for the thermal metal (red), ${\mathbb{Z}}_2$ (green), and trivial (blue) phase. The machine is highly confident of each phase but confused at the boundary. (b) The parameter $\mu$ of 1000 training data with system size $14\times 14$ is uniformly distributed within I: $[0.0,0.3]$, II: $[1.0,2.5]$, and III: $[4.0,10.0]$. During the training scheme, the network is tested by the data generated along $\mu \in [0.0,10.0]$ in the clean limit, resulting in accuracy over 90%. (c) Enlargement of the region surrounded by the orange dotted line in (a). (d) The averaged inputs $\langle P\left(\mathbf{r}\right)\rangle$ for $N_r=500$ in the vicinity of the boundaries. The parameters $(\mu ,W)$ are given as ${\text{X}}_1$: $(3,11.5),{\text{X}}_2$: $(3,10.75),{\text{Y}}_1$: $(5.25,10.5),{\text{Y}}_2$: $(5.5,10)$.

Figure 4

Average output of 200 binary-classifying ANNs trained with the clean-limit data for $t=1,\mathrm{\Delta }=3,{\mathrm{\Delta }}_2=0$. The parameter $\mu$ of 1000 training data with system size $14\times 14$ is uniformly distributed within (a) I: $[0.5,3.5]$ and II: $[6.0,10.0]$, (b) I, II, and III: $\mu =0.0,4.0$, each corresponding to the ${\mathbb{Z}}_2$ (green), the trivial (blue) phase, and the critical point (red). The performance of the machine is monitored with the test data generated at $\mu \in [0.0,10.0]$ in the clean limit, resulting in over 95% accuracy. The outputs above 0.75 for $\langle P\left(\mathbf{r}\right)\rangle$ with $N_r=500$ are indicated by the depth of the color, and merely gray for below 0.75.

Figure 5

Finite-size scaling of the dimensionless localization length, $\mathrm{\Lambda }$, by the transfer matrix method for the Hamiltonian defined by Eqs. (1, 2, 3, 4, 5) in the main text is shown in (a)–(c). The parameters are given as $(W,\mathrm{\Delta },{\mathrm{\Delta }}_2)$ = (a) $(5,3,0)$ under OBC, (b) $(12.5,3,0)$ under PBC, and (c) $(5,3,2)$ under OBC. Presented in (d)–(f) are the corresponding data collapses. The system width varies as $S=8,12,16$.

Figure 6

(a) and (d) ${\lambda }_1-{\lambda }_2$ as a function of the chemical potential $\mu$ and the disorder strength $W$ for class DIII Hamiltonian given by Eq. (1, 2, 3, 4, 5) of the main text. The parameters are taken as $(t,\mathrm{\Delta },{\mathrm{\Delta }}_2)$ = (a) (1, 3, 0) and (d) (1, 3, 2), respectively, and the system size is $20\times 20$. (b), (c), (e), and (f) show $\mu$ and $W$ dependencies of the eigenvalues ${\lambda }_i, i=1,2,...,7$, of the operator $\mathcal{A}$ for the system size $20\times 20$. The parameters are taken as $(t,\mathrm{\Delta },{\mathrm{\Delta }}_2,W)$ = (b) (1, 3, 0, 15), (c) (1, 3, 0, 5), (f) (1, 3, 2, 10), and (e) $(t,\mathrm{\Delta },{\mathrm{\Delta }}_2,\mu )=\text{(1,}\text{3,}\text{2,}\text{2)}$, respectively. The gray bars in (b), (c), (e), and (f) denote marginal areas.

Figure 7

(a) The output of ANN for single-shot $P\left(\mathbf{r}\right)$ for $\mathrm{\Delta }=3,{\mathrm{\Delta }}_2=2$. Boundaries between phases are hardly recognizable. (b) The single output for $P\left(\mathbf{r}\right)$, the average of 200 outputs for independently generated $P\left(\mathbf{r}\right)$, and the single output for $\langle P\left(\mathbf{r}\right)\rangle$ with $N_r=500$ from the top. The amplitude of the random potential is fixed as $W=15$.