标题： 双曲伊辛模型中边界敏感相的出现 显示英文标题

标题： Emergence of a Boundary-Sensitive Phase in Hyperbolic Ising Models

摘要： 在双曲格子上定义的物理系统可能表现出仅由于负曲率而出现的物质相。 我们关注的是在开放边界条件下伊辛模型的情况，并表明除了标准（高温）顺磁相和（低温）铁磁相之外，还会出现一个“中间”相。 在进行克朗默斯-瓦纳尔对偶性变换时，它会改变边界条件这一事实变得至关重要，因为有一部分格点位于边界上。 我们建议通过其对边界条件的敏感性来表征这个“中间”相，在此相中，体内的有序不是自发的，而是由边界效应引起的，这使其不同于朗道的自发对称性破缺范式。 通过开发一种限制了$\mathbb{Z}_2$对称性的角转移矩阵重正化群方法的扩展，我们提供了所有三种不同相及其对应的两阶段相变存在的数值证据，从而确立了完整的相图。 我们还确定了（自发的）中间相到铁磁相和（诱导的）顺磁相到中间相的转变点如何通过克朗默斯-瓦纳尔对偶关系相关联。 显示英文摘要

摘要： Physical systems defined on hyperbolic lattices may exhibit phases of matter that only emerge due to negative curvature. We focus on the case of the Ising model under open boundary conditions and show that an ``intermediate'' phase emerges in addition to standard (high-temperature) paramagnetic and (low-temperature) ferromagnetic phases. When performing the Kramers-Wannier duality the fact that it alters boundary conditions becomes crucial, since a finite fraction of lattice sites lie on the boundary. We propose to characterize this ``intermediate'' phase by its sensitivity to boundary conditions, wherein bulk ordering is not spontaneous but rather induced by boundary effects, setting it apart from the Landau paradigm of spontaneous symmetry breaking. By developing a $\mathbb{Z}_2$ symmetry restricted extension of the Corner Transfer Matrix Renormalization Group method, we provide numerical evidence for the existence of all three distinct phases and their corresponding two-stage phase transitions, thereby establishing the complete phase diagram. We also establish how the (spontaneous) intermediate-to-ferromagnetic and the (induced) paramagnetic-to-intermediate transition points are related by the Kramers-Wannier duality relation.