发表机构
School of Nuclear Science and Technology, University of Chinese Academy of Sciences; RIKEN Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS); Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), WPI, The University of Tokyo; Institute for Physics of Intelligence, The University of Tokyo（中国科学院大学核科学与技术学院; 理化学研究所交叉理论数学科学中心(iTHEMS); 东京大学宇宙物理数学研究所(Kavli IPMU，WPI); 东京大学智能物理学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出用两近邻法估计的固有维数作为几何探针，可分辨格点系统的多重相变，在q态时钟模型、U(1)希格斯模型中验证了其有效性，为全QCD的相关研究提供了方法支撑。
AI 中文摘要
具有多个邻近相变的格点系统面临两个相关挑战：分辨不同的相变尺度，以及识别与每个相变主要相关的自由度。我们证明，通过两近邻法估计的蒙特卡洛构型系综的固有维数，可为这两个问题提供几何诊断。在二维q态时钟模型中，固有维数能区分有序、准临界和无序区域，适用于间距较大（q=9）和间距较近（q=5）的Berezinskii-Kosterlitz-Thouless相变。对于q=5的情况，即使能量和磁化强度无法单独分辨这两个相变，中间相也会呈现为一个宽的低维谷。在四维U(1)希格斯模型中，我们引入基于规范不变 plaquette（面片）和希格斯变量的通道分解固有维数。每个通道的主导响应跟踪与相应自由度相关的相变，而通道组合则保留两者的特征。我们进一步证明，直接在规范可变场上评估的固有维数由规范轨道方向主导，这表明在解释构型空间几何前去除规范冗余的重要性。这些结果确立了通道分解固有维数作为探测具有多重相变的格点系统的几何探针，并推动其在全QCD中分离退禁闭和手征交叉尺度的应用。
英文摘要
Lattice systems with multiple nearby transitions pose two related challenges: resolving distinct transition scales and identifying the degrees of freedom primarily associated with each transition. We show that the intrinsic dimension of Monte Carlo configuration ensembles, estimated by the two-nearest-neighbors method, provides a geometric diagnostic for both problems. In the two-dimensional $q$-state clock model, the intrinsic dimension distinguishes the ordered, quasi-critical, and disordered regimes for both well-separated ($q=9$) and closely spaced ($q=5$) Berezinskii--Kosterlitz--Thouless transitions. In the $q=5$ case, the intermediate phase appears as a broad low-dimensional valley even when energy and magnetization do not separately resolve the two transitions. In the four-dimensional $U(1)$ Higgs model, we introduce channel-decomposed intrinsic dimensions based on gauge-invariant plaquette and Higgs variables. The dominant response of each channel tracks transitions associated with the corresponding degrees of freedom, while the combined channel retains features of both. We further show that intrinsic dimensions evaluated directly on gauge-variant fields are dominated by gauge-orbit directions, demonstrating the importance of removing gauge redundancy before interpreting configuration-space geometry. These results establish channel-decomposed intrinsic dimension as a geometric probe of lattice systems with multiple transitions and motivate its application to disentangling deconfinement and chiral crossover scales in full QCD.
Comments18 pages, 15 figures
