Quantitative and interpretable order parameters for phase transitions from persistent homology

We apply modern methods in computational topology to the task of discovering and characterizing phase transitions. As illustrations, we apply our method to four two-dimensional lattice spin models: the Ising, square ice, XY, and fully frustrated XY models. In particular, we use persistent homology, which computes the births and deaths of individual topological features as a coarse-graining scale or sublevel threshold is increased, to summarize multiscale and high-point correlations in a spin configuration. We employ vector representations of this information called persistence images to formulate and perform the statistical task of distinguishing phases. For the models we consider, a simple logistic regression on these images is sufficient to identify the phase transition. Interpretable order parameters are then read from the weights of the regression. This method suffices to identify magnetization, frustration, and vortex-antivortex structure as relevant features for phase transitions in our models. We also define persistence critical exponents and study how they are related to those critical exponents usually considered.

Automated summary

This research applies modern methods in computational topology to discover and characterize phase transitions, utilizing persistent homology to compute topological features and persistence images to distinguish phases. By employing logistic regression on these images, relevant features such as magnetization, frustration, and vortex-antivortex structure are identified as key factors in phase transitions. This method also defines persistence critical exponents and explores their relationship with traditionally considered critical exponents.