Abstract
Quantum many-body systems are characterized by patterns of correlations that define highly-non trivial manifolds when interpreted as data structures. Physical properties of phases and phase transitions are typically retrieved via simple correlation functions, that are related to observable response functions. Recent experiments have demonstrated capabilities to fully characterize quantum many-body systems via wave-function 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 from nearest neighbors (NN): the former is related to dataset 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 {\it 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 (non-commuting) 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.
Highlights
The path-integral (PI) formulation of quantum partition functions is arguably one of the most basic concepts in quantum many-body theory [1,2,3]
On a more general, yet abstract ground, the path integral of a many-body problem can be construed as a very complex multidimensional manifold embedded in a space that describes both spatial and imaginary-time coordinates: within this geometrical interpretation, while low-order properties of the PI manifold are in general well understood, it is currently unclear if global properties of such data structures can be informative about physical phenomena at all, or if they are constrained by universal properties of the many-body dynamics [3]
In a previous work [25], some of us have shown how classical partition functions exhibit very specific patterns in data space, that are universal in the vicinity of criticality, and where the corresponding structural transition can be understood utilizing simple arguments based on correlation functions
Summary
The path-integral (PI) formulation of quantum partition functions is arguably one of the most basic concepts in quantum many-body theory [1,2,3]. For all the models considered here, the intrinsic dimension displays a minimum at transition points—indicating that the geometry of the PI manifold simplifies at criticality This observation points to the fact that the PI manifold behaves independently from quantum correlations such as bipartite entanglement, that are often maximal at quantum critical points (QCPs) [5,30]; our findings are instead suggestive of the fact that the PI data structure is inheriting simplicity from the fact that the low-energy properties are captured by very few degrees of freedom (DOF)—and constrain the PI structure. The relation between PI data structure and universal behavior becomes apparent when performing a finite-size scaling (FSS) analysis of the intrinsic dimension [8]: the latter displays universal scaling collapse, whose functional form is dictated by the universality class (second-order or BKT), and by its critical exponent ν
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