On the criticality of the configuration-space statistical geometry

TL;DR

This paper introduces a geometric framework in configuration space to analyze phase transitions, linking it to real-space observables and demonstrating universal critical scaling through simulations and information geometry.

Contribution

It establishes a novel geometric perspective on quantum criticality by connecting configuration space statistics with phase transition diagnostics and universal scaling laws.

Findings

Universal scaling law in configuration space: $ ext{StdDev}(r_H) o L^{-2eta/ u}$

Fisher information on configuration space detects phase transitions regardless of measurement basis

Parity index from $P(r_H)$ characterizes topological phase transitions in SSH-Hubbard model

Abstract

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 $r_H$ 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 $\sqrt{\mathrm{Var}(r_H)}\sim L^{-2\beta/\nu}$, provided that the system possesses zero magnetization and satisfies $4\beta/\nu < d$. 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 $P(r_H)$ 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 $P(r_H)$ 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.