This paper systematically reviews a series of works of our research group in the new paradigm of phase transition research, namely “statistical geometry of configuration space”. This method identifies phase transitions from the raw data in an unsupervised manner and extracts universal critical behaviors by analyzing the distance distribution between microscopic configurations sampled by Monte Carlo. We demonstrate how this paradigm has evolved from a numerical exploration of complex models to a universal empirical scaling law, and, finally, to an analytical theoretical framework that connects geometric quantities in configuration space with correlation functions in real space. This work reveals how macroscopic criticality is encoded in the geometry of microscopic configurations, providing a brand-new, “white-box” and analytically tractable data-driven perspective for studying complex phase transition problems with unknown order parameters.
This paper systematically reviews a series of works of our research group in the new paradigm of phase transition research, namely “statistical geometry of configuration space”. This method identifies phase transitions from the raw data in an unsupervised manner and extracts universal critical behaviors by analyzing the distance distribution between microscopic configurations sampled by Monte Carlo. We demonstrate how this paradigm has evolved from a numerical exploration of complex models to a universal empirical scaling law, and, finally, to an analytical theoretical framework that connects geometric quantities in configuration space with correlation functions in real space. This work reveals how macroscopic criticality is encoded in the geometry of microscopic configurations, providing a brand-new, “white-box” and analytically tractable data-driven perspective for studying complex phase transition problems with unknown order parameters.
While phases and phase transitions are conventionally described by local order parameters in real space, we present a unified framework characterizing the phase transition through the geometry of configuration space defined by the statistics of pairwise distances rH between configurations. Focusing on the concrete example of Ising spins, we establish crucial analytical links between this geometry and fundamental real-space observables, i.e., the magnetization and two-point spin correlation functions. This link unveils the universal scaling law in the configuration space: the standard deviation of the normalized distances exhibits universal criticality as Var(rH)∼L−2β/ν, provided that the system possesses zero magnetization and satisfies 4β/ν<d. We validate this scaling with stochastic series expansion quantum Monte Carlo simulations of the transverse-field Ising model(TFIM). Furthermore, we propose configuration-space diagnostics that go beyond local real-space observables. First, the distribution probability P(rH) parameterized by the transverse field h forms a one-dimensional manifold. Information-geometric analyses, particularly the Fisher information defined on this manifold, successfully pinpoint the TFIM phase transition, regardless of the measurement basis. Second, for the Su-Schrieffer-Heeger Heisenberg model, a parity index derived from P(rH) successfully characterizes the symmetry-protected topological phase and its transition. Our work establishes configuration space geometry as a novel perspective on quantum criticality, revealing how macroscopic universal phenomena are encoded within its global statistical features.
In principle, the probability of configurations, determined by the system's partition function or wave function, encapsulates essential information about phases and phase transitions. Despite the exponentially large configuration space, we show that the generic correlation of distances between configurations, with a degree of freedom proportional to the lattice size, can probe phase transitions using importance sampling procedures like Monte Carlo simulations. The distribution of sampled distances varies significantly across different phases, suggesting universal critical behavior for uncertainty and participation entropy. For various classical spin models with different phases and transitions, finite-size analysis based on these quantities accurately identifies phase transitions and critical points. Notably, in all cases, the critical exponent derived from the uncertainty of distances equals the anomalous dimension governing real-space correlation decay. Thus, configuration space correlations, defined by distance uncertainties, share the same decay ratio as real-space correlations, determining the universality class of phase transitions. This work applies to diverse lattice models with different local degrees of freedom, e.g., two levels for Ising-like models, discrete multilevels for q-state clock models, and continuous local levels for the XY model, offering a robust, alternative method for understanding complex phases and transitions.
The identification of universal properties from minimally processed data sets is one goal of machine learning techniques applied to statistical physics.Here, we study how the minimum number of variables needed to accurately describe the important features of a data set-the intrinsic dimension (I d )-behaves in the vicinity of phase transitions.We employ state-of-the-art nearest-neighbors-based I d estimators to compute the I d of raw Monte Carlo thermal configurations across different phase transitions: first-order, second-order, and Berezinskii-Kosterlitz-Thouless.For all the considered cases, we find that the I d uniquely characterizes the transition regime.The finite-size analysis of the I d allows us to not only identify critical points with an accuracy comparable to methods that rely on a priori identification of order parameters but also to determine the corresponding (critical) exponent ν in the case of continuous transitions.For the case of topological transitions, this analysis overcomes the reported limitations affecting other unsupervised learning methods.Our work reveals how raw data sets display unique signatures of universal behavior in the absence of any dimensional reduction scheme and suggest direct parallelism between conventional order parameters in real space and the intrinsic dimension in the data space.
Critical phenomena, characterized by abrupt changes in macroscopic properties near phase transition points, exhibit universal scaling behavior independent of microscopic details. This paper explores the hypothesis that entropy serves as a fundamental universal scaling law governing these phenomena. We review the theoretical foundations of critical phenomena, including Landau theory and the renormalization group, highlighting the role of critical exponents and scaling functions. We then propose a framework where the configurational entropy, entanglement entropy, and other entropic measures exhibit universal scaling, linking them directly to the underlying critical exponents. Our methodology involves analyzing how entropy behaves under coarse-graining transformations near critical points, identifying a potential entropic fixed point in the renormalization group flow. We present theoretical arguments suggesting that the information content, as quantified by various entropy definitions, reflects the diverging correlation length and the emergent scale invariance at criticality. The discussion extends to how different entropic forms, such as Rényi and Tsallis entropies, might provide a more generalized framework for understanding critical behavior in complex and non-extensive systems. By demonstrating entropy's role as a universal scaling law, this work offers a unifying perspective on critical phenomena, deepening our understanding of phase transitions and the emergence of macroscopic order from microscopic fluctuations.
This paper provides a comprehensive review and synthesis of phase transitions and criticality in random linear systems. Such systems, characterized by matrices with stochastic entries, are ubiquitous across statistical physics, machine learning, signal processing, and control theory. We examine how their macroscopic properties undergo abrupt qualitative changes as underlying parameters vary, analogous to physical phase transitions. Criticality, the state at these phase boundaries, exhibits unique statistical properties like scaling laws and universal behavior, often defying characterization by conventional perturbation theories. The paper reviews foundational concepts from Random Matrix Theory and statistical mechanics of disordered systems, connecting them to modern high-dimensional data analysis and artificial intelligence. We discuss theoretical frameworks, including the replica and cavity methods, and numerical approaches employed to uncover these transitions. Furthermore, we highlight the practical implications of understanding criticality for designing robust algorithms, predicting system performance limits, and comprehending emergent complexity. Our aim is to offer an interdisciplinary synthesis, bridging classical insights with cutting-edge developments, to provide a unified perspective on these profound phenomena across diverse research fields.
Quantum many-body systems are characterized by patterns of correlations defining highly nontrivial manifolds when interpreted as data structures.Physical properties of phases and phase transitions are typically retrieved via correlation functions, that are related to observable response functions.Recent experiments have demonstrated capabilities to fully characterize quantum many-body systems via wavefunction snapshots, opening new possibilities to analyze quantum phenomena.Here, we introduce a method to data mine the correlation structure of quantum partition functions via their path integral (or equivalently, stochastic series expansion) manifold.We characterize path-integral manifolds generated via state-of-the-art quantum Monte Carlo methods utilizing the intrinsic dimension (ID) and the variance of distances between nearest-neighbor (NN) configurations: the former is related to data-set complexity, while the latter is able to diagnose connectivity features of points in configuration space.We show how these properties feature universal patterns in the vicinity of quantum criticality, that reveal how data structures simplify systematically at quantum phase transitions.This is further reflected by the fact that both ID and variance of NN distances exhibit universal scaling behavior in the vicinity of second-order and Berezinskii-Kosterlitz-Thouless critical points.Finally, we show how non-Abelian symmetries dramatically influence quantum data sets, due to the nature of (noncommuting) conserved charges in the quantum case.Complementary to neural-network representations, our approach represents a first elementary step towards a systematic characterization of path-integral manifolds before any dimensional reduction is taken, that is informative about universal behavior and complexity, and can find immediate application to both experiments and Monte Carlo simulations.