Engineering topological phases guided by statistical and machine learning methods

Thomas Mertz,Roser Valentí

doi:10.1103/physrevresearch.3.013132

Abstract

The search for materials with topological properties is an ongoing effort. In this article we propose a systematic statistical method, supported by machine learning techniques, that is capable of constructing topological models for a generic lattice without prior knowledge of the phase diagram. By sampling tight-binding parameter vectors from a random distribution, we obtain data sets that we label with the corresponding topological index. This labeled data is then analyzed to extract those parameters most relevant for the topological classification and to find their most likely values. We find that the marginal distributions of the parameters already define a topological model. Additional information is hidden in correlations between parameters. Here we present as a proof of concept the prediction of the Haldane model as the prototypical topological insulator for the honeycomb lattice in Altland-Zirnbauer (AZ) class A. The algorithm is straightforwardly applicable to any other AZ class or lattice, and could be generalized to interacting systems.4 MoreReceived 16 September 2020Accepted 12 January 2021DOI:https://doi.org/10.1103/PhysRevResearch.3.013132Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Physical SystemsChern insulatorsTopological insulatorsTechniquesMachine learningMonte Carlo methodsTight-binding modelCondensed Matter, Materials & Applied Physics

Highlights

In recent years machine learning techniques have enjoyed growing attention among the physics community
We compare the PDFs within the four different classes of hopping parameters in terms of DB(pt1,i, pt1,j ) etc., and observe that all distributions are very similar, except the ones of t2A and t2B. This observation lends itself as an argument for introducing a symmetry between the hoppings with equal PDFs. Taking into account this symmetry of the probability density functions and the correlations between features, we reduce the model to a six-parameter model with m, t1, t2A, t2B, t3, t4, which corresponds to 11 real features instead of the general 37
We have presented a scheme to learn the characteristics of topological phases and extract minimal models for a specific lattice

Summary

INTRODUCTION

In recent years machine learning techniques have enjoyed growing attention among the physics community. Dissecting first the well-known Haldane model [27] to benchmark and validate our findings, we look at the most general model on a honeycomb lattice and use our analysis to extract a topological prototype model for each individual class label.

DATA GENERATION

STATISTICAL METHOD

BENCHMARK CASE

GENERAL HONEYCOMB LATTICE

CONCLUSION & OUTLOOK