Unsupervised Classification of Topological Insulators

Abstract

Topological insulators are gapped states of quantum matter which cannot be adiabatically connected to conventional insulators [1] and are characterized, among other aspects, by gap less boundary modes. They constitute an active field of research and have the potential to be used for new technologies such as electronic devices with low power consumption, topological quantum computers, etc. While theorists have performed systematic classifications of such topological phases (for example, [2]), the identification of the precise topological character istics of a given Hamiltonian can still be challenging. Recently, machine learning algorithms have also been employed for “learning” phases and phase transitions in condensed matter systems [3]. However, most of these use supervised algorithms which require a-priori la belling of data sets. Our focus is on unsupervised algorithms, which have recently been used for the classification of topological phases [4, 5]. The advantage of unsupervised algorithms is that they don’t require labelled data. Such algorithms try to extract patterns from the data sets and classify the data into groups. This opens up the possibility of the algorithm to discover new phases or uncover patterns which haven’t been observed before, from raw data. We use the Diffusion Map algorithm to perform topological classification of 1-D 2 band Hamiltonians, and derive the corresponding auxiliary quantum many-body Hamiltonian for this problem.