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.
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.
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.
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.
Topology plays a fundamental role in our understanding of many-body physics, from vortices and solitons in classical field theory to phases and excitations in quantum matter. Topological phenomena are intimately connected to the distribution of information content that, differently from ordinary matter, is now governed by nonlocal degrees of freedom. However, a precise characterization of how topological effects govern the complexity of a many-body state, i.e., its partition function, is presently unclear. In this paper, we show how topology and complexity are directly intertwined concepts in the context of classical statistical mechanics. We concretely present a theory that shows how the Kolmogorov complexity of a classical partition function sampling carries unique, distinctive features depending on the presence of topological excitations in the system. We confront two-dimensional Ising, Heisenberg, and XY models on several topologies and study the corresponding samplings as high-dimensional manifolds in configuration space, quantifying their complexity via the intrinsic dimension. While for the Ising and Heisenberg models the intrinsic dimension is independent of the real-space topology, for the XY model it depends crucially on temperature: across the Berezkinskii-Kosterlitz-Thouless (BKT) transition, complexity becomes topology dependent. In the BKT phase, it displays a characteristic dependence on the homology of the real-space manifold, and, for g-torii, it follows a scaling that is solely genus dependent. We argue that this behavior is intimately connected to the emergence of an order parameter in data space, the conditional connectivity, which displays scaling behavior. Our approach paves the way for an understanding of topological phenomena emergent from many-body interactions from the perspective of Kolmogorov complexity.
Analog quantum simulators natively return large collections of single-shot projective measurements, the wave-function snapshots. These are high-dimensional configurations drawn from the Born distribution of the state they prepare. These collections carry far more information than is retained when they are simply compressed into low-order observables fixed in advance. Moreover, this procedure requires knowing beforehand what one is looking for. This thesis chooses to take a different route, asking what structure a dataset of snapshots possesses as a set of points in configuration space, and what kind of physics that structure reflects. The tools used here are drawn from un supervised learning and network theory, namely principal component analysis (PCA), intrinsic-dimension estimators (TWO-NN and BID), and the wave-function networks obtained by promoting snapshots to nodes and activating links according to the met ric structure of configuration space. They are chosen under two constraints, namely interpretability at every step, and the smallest possible amount of physical input: no order parameter, no symmetry, and no knowledge of the Hamiltonian. At equilibrium, wave-function networks are used to build a classification of one dimensional quantum phases based on snapshots alone. The scheme first selects, in the spirit of Occam’s razor, the measurement basis carrying the minimal information content, and then analyses the networks constructed in that basis. Paramagnetic and symmetry-broken phases display low complexity and “probability” networks, symmetry protected topological phases high complexity and Erdős–Rényi networks, and critical points high complexity and scale-free networks. Interestingly, gapped and gapless mat ter are treated on the same footing, and real-space decimation enters the scheme to resolve massive phases rather than flowing towards criticality. Out of equilibrium, the dynamical exponent governing transport of a conserved quan tity is shown to be imprinted on the singular-value spectrum of the snapshot matrix. In particular, it is shown that the PCA entropy, and in fact the leading eigenvalue alone, grows with the same exponent as the underlying spin or energy transport, at modest system sizes, evolution times and sample numbers. Ergodicity breaking is then ad dressed through the binary intrinsic dimension (BID) of the snapshots, in the quantum sun model: the BID grows and saturates at the Thouless time, reaching its maximum in the ergodic phase, remaining system-size independent in the non-ergodic one, and growing sub-maximally at criticality, where the data occupy an extensive but lower dimensional manifold. The associated networks reproduce the equilibrium dichotomy, being random in the ergodic phase and scale-free at the transition, whose location can be identified by the BID within a few percent of spectral estimates. Preliminary experimental observations, not reported here, show analogous features. Taken together, these results indicate that datasets orders of magnitude smaller than the Hilbert-space dimension suffice to identify collective properties of quantum matter through the study of the structure of the data points in configuration space. Moreover, the same structural classes recur across ground states and quenched dynamics, with no order parameter supplied and under realistic experimental imperfections. The final chapter elaborates on another aspect of quantum simulator capabilities, that is, that to realize gauge theory dynamics. There, one-dimensional U(1) lattice gauge theories with a topological θ term are mapped exactly onto kinetically con strained fermions in an emergent Stark potential. The mapping accounts for the slow relaxation reported in recent Rydberg experiments: prethermalization near an inte grable point at θ = 0, and Stark localization with Hilbert-space fragmentation at θ ̸ =0.
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.
Modern quantum experiments can probe many-body wavefunctions through projective measurements, providing snapshots of individual many-body configurations. A fundamental question is whether the intrinsic structure of their measurement distributions can reveal quantum dynamics beyond predefined observables. Here, we address this question at the many-body ergodicity-breaking transition using two complementary characteristics of nonequilibrium wavefunction snapshots: their binary intrinsic dimension (BID) and the topology of wavefunction networks constructed from Hamming distances. We show that the critical point is characterized by an extensive but submaximal BID, and scale-free network connectivity with highly connected hubs. These signatures distinguish the ergodic, critical, and nonergodic regimes, and provide experimentally accessible probes of ergodicity breaking from projective measurement snapshots.