Phase diagram of a disordered higher-order topological insulator: A machine learning study

A higher-order topological insulator is a new concept of topological states of matter, which is characterized by the emergent boundary states whose dimensionality is lower by more than two compared with that of the bulk, and draws a considerable interest. Yet, its robustness against disorders is still unclear. In this work, we investigate a phase diagram of higher-order topological insulator phases in a breathing kagome model in the presence of disorders by using a state-of-the-art machine learning technique. We find that the corner states survive against the finite strength of disorder potential as long as the energy gap is not closed, indicating the stability of the higher-order topological phases against the disorders.

Figure 1

A kagome flake with (a) a rhombus geometry and (d) a triangle geometry. In each geometry, corners are denoted by green circles. The energy spectrum of the rhombus as a function of $t_1/t_2$ for (b) a rhombus geometry and (e) a triangle geometry. The energy spectrum with the disorder $W=1.1$ for (c) a rhombus geometry and (f) a triangle geometry.

Figure 2

The architecture of the neural network. The first two layers are two-dimensional convolutional layers and the rests are linear layers. We use some max pooling layers and batch normalization. The size of the layers are shown in the figure.

Figure 3

The modification of the kagome lattice to the square lattice for a rhombus geometry (upper) and a triangle geometry (lower). We add empty sites denoted by gray circles.

Figure 4

Wave functions on a rhombus geometry. (a) The averaged wave functions of the highest occupied state with $W=1$. The average is taken over 30 samples. (b) The single-shot wave functions of the highest occupied state with $W=1$.

Figure 5

Wave functions on a triangle geometry. (a) The averaged wave functions of the highest occupied state with $W=1$. The average is taken over 30 samples. (b) The single-shot wave functions of the highest occupied state with $W=1$. The average over three corner states is also taken (see the main text).

Figure 6

The phase diagram of the breathing kagome lattice with disorders for the rhombus geometry. (a) The phase diagrams is plotted for the averaged wave functions. The blue, right-blue, yellow, and orange areas denote trivial, HOTI 1, HOTI 2, and metallic phases, respectively. Green dashed lines denote the energy gap between the corner states and the bulk/edge states. (b) The phase diagram is plotted by the averaged probabilities for 30 single-shot wave functions.

Figure 7

The phase diagram of the breathing kagome lattice with disorders for the triangle geometry. (a) The phase diagrams is plotted for the averaged wave functions. The red, blue and green areas denote trivial, HOTI, and metallic phases, respectively. (b) The phase diagram is plotted by the averaged probabilities for 30 single-shot wave functions.

Figure 8

(a) The IPR for the averaged wave functions. (b) The averaged IPR for the single-shot wave functions. The number of the samples are 30 for each parameters. The system size $M=10$. Green dashed lines are same to the lines in Fig. 6. These figures are plotted on a logarithmic scale.