While phases and phase transitions are conventionally described by local order parameters in real space, we present a unified framework that characterizes phase transitions 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 $\sqrt{\mathrm{Var}\left(r_H\right)}\sim L^{-2\beta /\nu }$, provided that the system 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 $P\left(r_H\right)$ parametrized 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 $P\left(r_H\right)$ 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.

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• Phase transitions
• Lattice models in statistical physics
• Critical phenomena
• Information theory
• Monte Carlo methods