On the criticality of the configuration-space statistical geometry

TL;DR

The paper develops a configuration-space geometry framework to study quantum and classical phase transitions by examining the distribution of pairwise configuration distances $r_H$. A central result is the universal scaling $\sqrt{Var(r_H)} \sim L^{-2\beta/\nu}$ at criticality under zero magnetization and $4\beta/\nu < d$, linking configuration-space statistics to real-space critical exponents, with validation in 1D and 2D TFIM and extensions to BKT-like regimes. Beyond local probes, the work introduces basis-independent Fisher information on the $P(r_H)$ manifold and a parity index for SPT phases, enabling detection of criticality and topological transitions in settings where traditional order parameters fail. Numerically, SSE QMC confirms the predicted scaling and demonstrates that nonlocal configuration-space features can reveal phase structure in both conventional and topological contexts, including SSH-like chains. The findings offer a global, basis-agnostic perspective on quantum criticality and suggest directions toward higher-order geometric invariants and ML-inspired analyses of configuration-space landscapes.

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 $\sqrt{\mathrm{Var}(r_H)}\sim L^{-2β/ν}$, 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 $P(r_H)$ 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.

Figures (14)

• Figure 1: (a) The evolution of probability distribution $P(r_{H})$ as a function of the transverse field $h$, for 1D TFIM with $L=192$. (b) The standard deviation $\sqrt{\textrm{Var}(r_H)}$ extracted from data statistics, depicted by empty markers; and the theoretically estimated value using real-space observables according to Eq. \ref{['eq:var_rH']}, depicted by solid lines. In both panels, the vertical dashed line depicts the critical point $h_c=1$.
• Figure 2: Finite-size scaling analysis for $\sqrt{\mathrm{Var}(r_H)}$ of 1D TFIM. (a) The double-logarithmic plot of $\sqrt{\mathrm{Var}(r_H)}$ versus system size $L$ for several values of the transverse field $h$ near the critical point. The solid line depicts the best power-law fitting $\sim L^{-1/4}$ at the critical point. (b) Data collapse according to the scaling ansatz in Eq. \ref{['eq:scaling_con']}. The optimal parameters obtained from minimizing the collapse error [Here and after, the data collapse follows the procedure in Appendix \ref{['app:scaling_details']}], and they agree with the known values for the 2D Ising universality class ($h_c=1$, $\nu=1$, $2\beta/\nu=1/4$).
• Figure 3: (a) The evolution of probability distribution $P(r_{H})$ as a function of the transverse field $h$, for 2D TFIM with $L=80$. (b) The standard deviation $\sqrt{\textrm{Var}(r_H)}$ extracted from data statistics, depicted by empty markers; and the theoretically estimated value using real-space observables according to Eq. \ref{['eq:var_rH']}, depicted by solid lines. In both panels, the vertical dashed line depicts the critical point $h_c=3.044$.
• Figure 4: Finite-size scaling analysis for $\sqrt{\mathrm{Var}(r_H)}$ of 2D TFIM. (a) The double-logarithmic plot of $\sqrt{\mathrm{Var}(r_H)}$ versus system size $L$ for several values of the transverse field $h$ near the critical point. The solid line depicts the best power-law fitting $\sim L^{-0.971}$ at $h=3.045$, which is the closest data point to the known $h_c$. (b) Data collapse according to the scaling ansatz in Eq. \ref{['eq:scaling_con']}. The optimal parameters obtained from minimizing the collapse error; the numerical value obtained critical point agrees well with the known values for the 3D Ising universality class ($h_c=3.044$); $\nu=0.666$ and $2\beta/\nu=0.932$ are slightly away from the theoretical value ($\nu=0.630$, $2\beta/\nu=1.036$).
• Figure 5: Finite-size scaling analysis for $\sqrt{\mathrm{Var}(r_H)}$ of 3D classical Ising model, i.e., TFIM with $h=0$. (a) The double-logarithmic plot of $\sqrt{\mathrm{Var}(r_H)}$ versus system size $L$ for several values of temperatures $T$ near the critical point. The solid line depicts the predicted critical decay $\sim L^{-1.036}$. (b) Data collapse according to the scaling ansatz in Eq. \ref{['eq:scaling_con']}. The optimal parameters are obtained by minimizing the collapse error.