Detecting topological phases in the square-octagon lattice with statistical methods

TL;DR

This paper introduces a feature engineering approach combined with statistical learning to identify topological phases in the square-octagon lattice, especially at high-order van Hove singularities, advancing the analysis of topological insulators.

Contribution

It extends existing statistical methods by incorporating feature engineering to better detect topological states in complex lattice systems.

Findings

Identified polynomial parameter combinations linked to topological phases.

Applied method to square-octagon lattice at van Hove singularity.

Demonstrated effectiveness in uncovering topological states.

Abstract

Electronic systems living on Archimedean lattices such as kagome and square-octagon networks are presently being intensively discussed for the possible realization of topological insulating phases. Coining the most interesting electronic topological states in an unbiased way is however not straightforward due to the large parameter space of possible Hamiltonians. A possible approach to tackle this problem is provided by a recently developed statistical learning method [T. Mertz and R. Valent\'i, Phys. Rev. Research 3, 013132 (2021)], based on the analysis of a large data sets of randomized tight-binding Hamiltonians labeled with a topological index. In this work, we complement this technique by introducing a feature engineering approach which helps identifying polynomial combinations of Hamiltonian parameters that are associated with non-trivial topological states. As a showcase, we employ this method to investigate the possible topological phases that can manifest on the square-octagon lattice, focusing on the case in which the Fermi level of the system lies at a high-order van Hove singularity, in analogy to recent studies of topological phases on the kagome lattice at the van Hove filling.