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 characterizing 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. We discuss a holographic correspondence between boundary and bulk behaviors and derive exact expressions for boundary correlation functions on Cayley trees.

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• Classical statistical mechanics
• Collective behavior in networks
• Critical phenomena
• Exotic phases of matter
• Gauge-gravity dualities
• Network phase transitions
• Phase diagrams
• Phase transitions
• Quantum information theory