标题： 关于配置空间统计几何的临界性 显示英文标题

标题： On the criticality of the configuration-space statistical geometry

摘要： 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 $4\beta/\nu < d$. Numerical stochastic series expansion quantum Monte Carlo simulations on the transverse-field Ising model (TFIM) validate this scaling law: (i) It is perfectly validated in the one-dimensional TFIM, where all theoretical criteria are satisfied; (ii) Its robustness is confirmed in the two-dimensional TFIM, where, despite the theoretical applicability condition being at its marginal limit, our method robustly captures the effective scaling dominated by physical correlations; (iii) The method's specificity is demonstrated via a critical control experiment in the orthogonal $\hat{\sigma}^x$ basis, where no long-range order exists, correctly reverts to a non-critical background scaling. Moreover, 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 measuring basis. 显示英文摘要

摘要： 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 $4\beta/\nu < d$. Numerical stochastic series expansion quantum Monte Carlo simulations on the transverse-field Ising model (TFIM) validate this scaling law: (i) It is perfectly validated in the one-dimensional TFIM, where all theoretical criteria are satisfied; (ii) Its robustness is confirmed in the two-dimensional TFIM, where, despite the theoretical applicability condition being at its marginal limit, our method robustly captures the effective scaling dominated by physical correlations; (iii) The method's specificity is demonstrated via a critical control experiment in the orthogonal $\hat{\sigma}^x$ basis, where no long-range order exists, correctly reverts to a non-critical background scaling. Moreover, 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 measuring basis.