Deep learning for disordered topological materials through entanglement spectrum

Calculation of topological invariants for crystalline systems is well understood in terms of the Wilson loop technique in reciprocal space, allowing for algorithmic evaluation of the invariants for a wide spectrum of materials. While this same approach may still be valid for disordered or complex materials, where the supercell must be big enough to capture the physics of the material, it becomes a limiting factor as the calculation is expensive, disabling high-throughput computations. On top of that, the Wilson loop technique is only well-defined in insulating materials, i.e. systems whose bands are fully occupied. There are several methods to try to overcome these difficulties, such as the local Chern marker or the Bott index, which are defined in real space, but that are only applicable without time-reversal symmetry. In our work, we present a new technique based on the entanglement spectrum of a system, which contains the topological nature as it measures charge pumping. We show that it is possible to train a neural network to distinguish between trivial and topological phases using the entanglement spectrum from crystalline phases, and use it to predict the topological phase diagram of time-reversal disordered systems, or fractal lattices such as the Bethe lattice. This approach is shown to be robust, as it does not depend on boundary conditions and can be computed without need of a gap, also providing a speed-up compared with the Wilson loop technique.