Real space mapping of topological invariants using artificial neural networks

TL;DR

This paper demonstrates that artificial neural networks can be trained to identify topological invariants locally in real space, enabling the characterization of topological phases without global wavefunction information.

Contribution

The authors introduce a neural network approach to evaluate topological invariants locally, applicable to large systems with spatially modulated parameters.

Findings

Neural networks accurately identify topological domains in real space.

The method predicts the location of in-gap states.

Local topological evaluation is feasible for large, non-translational systems.

Abstract

Topological invariants allow to characterize Hamiltonians, predicting the existence of topologically protected in-gap modes. Those invariants can be computed by tracing the evolution of the occupied wavefunctions under twisted boundary conditions. However, those procedures do not allow to calculate a topological invariant by evaluating the system locally, and thus require information about the wavefunctions in the whole system. Here we show that artificial neural networks can be trained to identify the topological order by evaluating a local projection of the density matrix. We demonstrate this for two different models, a 1-D topological superconductor and a 2-D quantum anomalous Hall state, both with spatially modulated parameters. Our neural network correctly identifies the different topological domains in real space, predicting the location of in-gap states. By combining a neural network with a calculation of the electronic states that uses the Kernel Polynomial Method, we show that the local evaluation of the invariant can be carried out by evaluating a local quantity, in particular for systems without translational symmetry consisting of tens of thousands of atoms. Our results show that supervised learning is an efficient methodology to characterize the local topology of a system.