# Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology

arXiv: https://arxiv.org/abs/2109.10960 · HTML: https://mlsanity.com/paper/2109.10960/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 59 references that are in the map.
- First posted on arXiv 22 Sep 2021.

## Builds on (most cited first, 12 of 12)

- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [giotto-tda: A Topological Data Analysis Toolkit for Machine Learning and Data Exploration](https://mlsanity.com/paper/2004.02551/) (2020)
- [Discovering Phase Transitions with Unsupervised Learning](https://mlsanity.com/paper/1606.00318/) (2016)
- [Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams](https://mlsanity.com/paper/1802.03569/) (2018)
- [Quantum field-theoretic machine learning](https://mlsanity.com/paper/2102.09449/) (2021)
- [The density of expected persistence diagrams and its kernel based estimation](https://mlsanity.com/paper/1802.10457/) (2018)
- [Probing hidden spin order with interpretable machine learning](https://mlsanity.com/paper/1804.08557/) (2018)
- [Unveiling phase transitions with machine learning](https://mlsanity.com/paper/1904.01486/) (2019)
- [Extending machine learning classification capabilities with histogram reweighting](https://mlsanity.com/paper/2004.14341/) (2020)
- [Adding machine learning within Hamiltonians: Renormalization group transformations, symmetry breaking and restoration](https://mlsanity.com/paper/2010.00054/) (2020)
- [Topological Persistence Machine of Phase Transitions](https://mlsanity.com/paper/2004.03169/) (2020)

## Built on by (most cited first, 1 of 1)

- [Spectral origin of the topological gap exponent d + η: mechanism, kernel, decomposition, and scope](https://mlsanity.com/paper/2609.09159/) (2026)

## Abstract (arXiv)

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.