Learning quantum phase transitions through Topological Data Analysis

A central subject in Condensed Matter Physics and Statistical Mechanics is the study of phase transitions and critical phenomena [1, 2]. In the last decades, due to the increasing computer power resources, numerical methods have become an indispensable tool for the analysis of classical and quantum interacting systems. Most of these methods, such as Monte Carlo simulations, are performed at finite size systems, which demand the analysis by scaling theories to avoid misleading finite size effects [3-6]. However, depending on the type of systems (classical or quantum), or the geometry/dimensionality, performing a finite size scaling (FSS) analysis may be a challenge - sometimes an unfeasible task -, due to technical bottlenecks: for instance, as a paradigm, in quantum Monte Carlo simulations the occurrence of the infamous minus-sign problem, i.e. the occurrence a negative statistical weight, restricts the simulations to small lattice sizes [7-9]. Another instance is the analysis of three-dimensional systems, in which an extrapolation to the thermodynamic limit is very demanding, even in absence of the sign problem. In view of this, it is worth developing techniques that could give hints of the existing phases and their phase transitions at finite small system sizes, but, at the same time, could also provide quantitatively reasonable critical points. With the advent of big data analysis, e.g. with machine learning techniques, a great expectation is placed to this end. Indeed, over the past few years, there has been an effort to develop and benchmark supervised and unsupervised machine learning techniques [10-12].