Intrinsic Dimension of Path Integrals: DataMining Quantum Criticality and Emergent Simplicity
Abstract
Quantum manybody 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 manybody 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 pathintegral manifolds generated via stateoftheart quantum Monte Carlo methods utilizing the intrinsic dimension (ID) and the variance of distances between nearestneighbor (NN) configurations: 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 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 secondorder and BerezinskiiKosterlitzThouless critical points. Finally, we show how nonAbelian symmetries dramatically influence quantum data sets, due to the nature of (noncommuting) conserved charges in the quantum case. Complementary to neuralnetwork representations, our approach represents a first elementary step towards a systematic characterization of pathintegral 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.
 Publication:

PRX Quantum
 Pub Date:
 August 2021
 DOI:
 10.1103/PRXQuantum.2.030332
 arXiv:
 arXiv:2103.02640
 Bibcode:
 2021PRXQ....2c0332M
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Condensed Matter  Disordered Systems and Neural Networks;
 Physics  Computational Physics;
 Physics  Data Analysis;
 Statistics and Probability
 EPrint:
 19 pages, 13 figures, version accepted for publication in Phys. Rev. X Quantum