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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.00787 |
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| _version_ | 1866913148753149952 |
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| author | Liu, Yu-Jing Su, Wen-Yu Yang, Yong-Feng Ma, Nvsen Cheng, Chen |
| author_facet | Liu, Yu-Jing Su, Wen-Yu Yang, Yong-Feng Ma, Nvsen Cheng, Chen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00787 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the criticality of the configuration-space statistical geometry Liu, Yu-Jing Su, Wen-Yu Yang, Yong-Feng Ma, Nvsen Cheng, Chen Statistical Mechanics 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. |
| title | On the criticality of the configuration-space statistical geometry |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2508.00787 |