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Topological Persistence Machine of Phase Transitions
(2020)
Quoc Hoan Tran, Mark Chen, Yoshihiko Hasegawa
Abstract
The study of phase transitions from experimental data becomes challenging, especially when little prior knowledge of the system is available. Topological data analysis is an emerging framework for characterizing the shape of data and has recently achieved success in detecting structural transitions in material science such as glass-liquid transition. However, data obtained from physical states may not have explicit shapes as structural materials. We propose a general framework called topological persistence machine to construct the shape of data from correlations in states; hence decipher phase transitions via the qualitative changes of the shape. Our framework enables an effective and unified approach in phase transition analysis. We demonstrate the impact in highly precise detection of Berezinskii-Kosterlitz-Thouless phase transitions in the classical XY model, and quantum phase transition in the transverse Ising model and Bose-Hubbard model. Intriguingly, these phase transitions have proven to be notoriously difficult in traditional methods but can be characterized in our framework without requiring prior knowledge about phases. Our approach is thus expected applicable and brings a remarkable perspective for exploring phases of experimental physical systems.
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Quantitative Analysis of Phase Transitions in Two-Dimensional XY Models Using Persistent Homology
(2022)
Nicholas Sale, Jeffrey Giansiracusa, Biagio Lucini
Abstract
We use persistent homology and persistence images as an observable of three different variants of the two-dimensional XY model in order to identify and study their phase transitions. We examine models with the classical XY action, a topological lattice action, and an action with an additional nematic term. In particular, we introduce a new way of computing the persistent homology of lattice spin model configurations and, by considering the fluctuations in the output of logistic regression and k-nearest neighbours models trained on persistence images, we develop a methodology to extract estimates of the critical temperature and the critical exponent of the correlation length. We put particular emphasis on finite-size scaling behaviour and producing estimates with quantifiable error. For each model we successfully identify its phase transition(s) and are able to get an accurate determination of the critical temperatures and critical exponents of the correlation length.