学习Potts模型与$Z_3$环面码：高阶与普通Nishimori临界性
Learning Potts Models and $Z_3$ Toric Codes: Higher and Ordinary Nishimori Criticality
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中文总结 AI 辅助
该研究在二维q态Potts模型学习相图中识别出高阶Nishimori临界点，分析其相图结构、普适量及RG流特性，并通过数值验证，还将结果推广至变形$\boldsymbol{\text{Z}_q}$环面码的信息临界性研究。
中文摘要 AI 辅助
受此前伊辛模型研究的启发，我们在键能测量下的二维q态Potts模型（2 < q ≤ 4）的学习相图中识别出一条“高阶”西岛线。这条“高阶”西岛线与Potts模型的临界温度线在一个“高阶”西岛临界点处相交，该点是有限推断强度下的三临界点，将顺磁相、铁磁相和“自旋玻璃”相分隔开。我们利用分析工具，讨论了该丰富相图的一般结构，其包含两个不稳定不动点和三个稳定不动点，并采用可进行精确计算的高斯测量协议，获得了包括爱德华兹-安德森关联函数衰减指数在内的多个普适量的精确结果。我们通过大量数值工具，针对通用的离散q态测量协议验证了这些结论，并确定了高阶和普通西岛临界点的位置以及各不动点之间的重整化群（RG）流的精确数值估计。我们还讨论了学习相图中临界点的卡西米尔有效中心电荷，以及它们沿测量诱导的RG流的单调“减小”，这一结论由c-有效定理及其推广在非微扰层面确立，并将其与键无序Potts模型中相应RG流的单调增加进行对比。最后，我们基于“埃利楚尔定理”讨论了一个一般性论点，该论点确立了普通西岛临界点在其各自学习相图中的稳定性。等效地，我们的结果描述了一个受监控的变形ℤ_q环面码，其中三临界“高阶”西岛点是一个“信息”临界点，将稳定量子相、经典相和无记忆相分隔开。
英文摘要
Motivated by a previous Ising study, we identify a ${\it higher}$ Nishimori line in the learning phase diagram of the $2D$ $q$-state Potts model $(2 < q\leq 4)$ under bond-energy measurements. This ${\it higher}$ Nishimori line meets the critical temperature line of the Potts model, in a ${\it higher}$ Nishimori critical point -- a tricritical point at finite inference strength that separates a paramagnetic, a ferromagnetic and a 'spin-glass' phase. With analytical tools, we discuss the general structure of the rich phase diagram, which contains two unstable and three stable fixed points, and obtain a number of exact results for universal quantities, including the decay exponent of the Edwards-Anderson correlator, using a Gaussian measurement protocol which allows for exact calculations. Using extensive numerical tools, we confirm these statements for a generic, discrete $q$-state measurement protocol and determine precise numerical estimates for the location of higher and ordinary Nishimori critical points as well as RG flows between the various fixed points. We also discuss the Casimir effective central charges of the critical points in the learning phase diagram, and their monotonic ${\it decrease}$ along measurement-induced RG flows, as established non-perturbatively by the c-effective theorem and its extensions, and contrast it to the monotonic increase along the corresponding RG flows in the random-bond Potts model. Finally, we discuss a general argument based on ${\it Elitzur's \; theorem}$ that establishes stability of the ordinary Nishimori critical points in their respective learning phase diagrams. Equivalently, our results describe a monitored deformed $\mathbb{Z}_q$ toric code where the tricritical ${\it higher}$ Nishimori point is an 'information' critical point that separates stable quantum, classical, and no memory phases.
