The study of phase transitions using data-driven approaches is 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 the glass-liquid transition. However, data obtained from physical states may not have explicit shapes as structural materials. We thus propose a general framework, termed "topological persistence machine," to construct the shape of data from correlations in states, so that we can subsequently decipher phase transitions via qualitative changes in the shape. Our framework enables an effective and unified approach in phase transition analysis. We demonstrate the efficacy of the approach in detecting the Berezinskii-Kosterlitz-Thouless phase transition in the classical XY model and quantum phase transitions in the transverse Ising and Bose-Hubbard models. Interestingly, while these phase transitions have proven to be notoriously difficult to analyze using traditional methods, they can be characterized through our framework without requiring prior knowledge of the phases. Our approach is thus expected to be widely applicable and will provide practical insights for exploring the phases of experimental physical systems.
The study of phase transitions using data-driven approaches is 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 the glass-liquid transition. However, data obtained from physical states may not have explicit shapes as structural materials. We thus propose a general framework, termed "topological persistence machine," to construct the shape of data from correlations in states, so that we can subsequently decipher phase transitions via qualitative changes in the shape. Our framework enables an effective and unified approach in phase transition analysis. We demonstrate the efficacy of the approach in detecting the Berezinskii-Kosterlitz-Thouless phase transition in the classical XY model and quantum phase transitions in the transverse Ising and Bose-Hubbard models. Interestingly, while these phase transitions have proven to be notoriously difficult to analyze using traditional methods, they can be characterized through our framework without requiring prior knowledge of the phases. Our approach is thus expected to be widely applicable and will provide practical insights for exploring the phases of experimental physical systems.
Topological data analysis (TDA), particularly persistent homology, has been proposed as a tool for detecting topological phase transitions in systems lacking conventional order parameters, such as the Berezinskii–Kosterlitz–Thouless (BKT) transition in the 2D XY model. We subject this proposal to rigorous empirical scrutiny, identifying and correcting two compounding numerical artifacts. First, allowing the spin-angle field to undergo an unbounded random walk during Metropolis updates introduces artificial scalar gradients that corrupt the sublevel-set filtration; this is corrected by explicitly wrapping the phase to (-π, π]. Second, and more critically, we identify that single-snapshot measurements at low temperatures capture transient, non-equilibrium vortex populations, systematically inflating the vortex baseline and distorting the inferred onset temperature and dynamic range. When both artifacts are corrected through proper measurement-phase averaging (30 snapshots per seed, n=15 independent seeds, L=32, 19-point temperature sweep over T∈[0.60,1.50]), the TDA observable is revealed to be not merely a weaker proxy for the BKT transition, but one that lacks the transition's defining signature entirely. The corrected analysis reverses the sign of our two principal quantitative claims. The onset temperatures, previously reported as statistically indistinguishable (ΔT₀ ≈ 0.01±0.05), are in fact separated by a 7σ gap: vortex counting detects the transition significantly earlier than TDA (T₀^Vortex = 1.196±0.002 vs. T₀^TDA = 1.294±0.014). The parametric curvature of TDA relative to vortex density, previously reported as statistically insignificant (1.62σ), is in fact a 6.3σ effect (+0.034±0.005). The dynamic-range disparity is widened from the originally reported ~75× vs. ~2.4× to ~228× vs. ~1.66×. We conclude that this specific TDA observable does not detect the BKT transition and cannot serve as a viable diagnostic for topological defect unbinding. These results constitute a definitive negative benchmark for future TDA applications in statistical mechanics.
Machine-learning driven models have proven to be powerful tools for the identification of phases of matter. In particular, unsupervised methods hold the promise to help discover new phases of matter without the need for any prior theoretical knowledge. While for phases characterized by a broken symmetry, the use of unsupervised methods has proven to be successful, topological phases without a local order parameter seem to be much harder to identify without supervision. Here, we use an unsupervised approach to identify boundaries of the topological phases. We train artificial neural nets to relate configurational data or measurement outcomes to quantities like temperature or tuning parameters in the Hamiltonian. The accuracy of these predictive models can then serve as an indicator for phase transitions. We successfully illustrate this approach on both the classical Ising gauge theory as well as on the quantum ground state of a generalized toric code.
Abstract Machine-learning driven models have proven to be powerful tools for the identification of phases of matter. In particular, unsupervised methods hold the promise to help discover new phases of matter without the need for any prior theoretical knowledge. While for phases characterized by a broken symmetry, the use of unsupervised methods has proven to be successful, topological phases without a local order parameter seem to be much harder to identify without supervision. Here, we use an unsupervised approach to identify boundaries of the topological phases. We train artificial neural nets to relate configurational data or measurement outcomes to quantities like temperature or tuning parameters in the Hamiltonian. The accuracy of these predictive models can then serve as an indicator for phase transitions. We successfully illustrate this approach on both the classical Ising gauge theory as well as on the quantum ground state of a generalized toric code.
The authors extend the recent successful application of neural networks fordistinguishing phases of matter to the case of topologically driven phase transitions. They demonstrate that supervised networks correctly identify the Kosterlitz-Thouless transition in the two-dimensional classical XY model. However, their analysis suggests that a naive classification of phases is based on bulk features, rather than on detecting local topological defects. While they show that it is possible to fine-tune a network to recognize vortices, it generally remains a difficult task to detect topological defects without $a\phantom{\rule{0.333em}{0ex}}p\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}i$ knowledge.
Identifying phase transitions in complex many-body systems traditionally necessitates the definition of specific order parameters, a task often requiring prior knowledge of the statistical model and the symmetry-breaking mechanism. In this work, we propose a framework for detecting phase transitions directly from raw (experimental) data without requiring knowledge of the underlying model Hamiltonian, parameters, or pre-defined labels. Inspired by generative modeling in machine learning, our method utilizes autoregressive networks to estimate the normalized probability distribution of the system from raw configuration data. We then quantify the intrinsic sensitivity of this learned distribution to control parameters (such as temperature) to construct a robust indicator of phase transitions. This indicator is based on the expectation of the change in absolute logarithmic probability, derived entirely from the raw data. Our approach is purely data-driven: it takes raw data across varying control parameters as input and outputs the most likely estimate of the phase transition point. To validate our approach, we conduct extensive numerical experiments on the 2D Ising model on both triangular and square lattices, and on the Sherrington-Kirkpatrick (SK) model utilizing raw data generated via Markov Chain Monte Carlo and Tensor Network methods. The results demonstrate that our generative approach accurately identifies phase transitions using only raw data. Our framework provides a general tool for exploring critical phenomena in model systems, with the potential to be extended to realistic experimental data where theoretical descriptions remain incomplete.
The identification and characterization of phase transitions is one of the oldest and most central problems in condensed matter physics, traditionally approached through order parameters, free-energy expansions, renormalization-group analysis, and direct numerical simulation. Over the past decade, machine learning (ML) has emerged as a complementary and, in some regimes, transformative methodology for this task. Supervised classifiers can learn to distinguish phases directly from raw configurations without an a priori order parameter; unsupervised and self-supervised techniques such as principal component analysis, autoencoders, and learning-by-confusion can locate transition points with minimal or no labeled data; and generative and variational architectures, including neural-network quantum states, now approximate ground-state wavefunctions of strongly correlated systems well beyond the reach of exact diagonalization. This paper reviews the theoretical foundations of phase transitions relevant to machine-learning applications, surveys the principal families of algorithms used to detect and classify phases, and examines their application across classical spin systems, percolation, topological and quantum phases, strongly correlated electron systems, and experimentally measured spectroscopic data. Representative case studies are discussed, including convolutional and graph-neural-network classifiers of lattice configurations, explainable-ML detection of thermodynamic transitions in photoemission spectra, and deep-learning-guided discovery of high-temperature superconductors. The review closes with a critical discussion of persistent challenges interpretability, data scarcity in correlated-electron experiments, generalization across Hamiltonians, and the reliability of learned order parameters near criticality and outlines directions for future research, including physics-informed architectures, uncertainty quantification, and closed-loop integration with autonomous experimentation.