Emergence of a Boundary-Sensitive Phase in Hyperbolic Ising Models

Abstract

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

19 pages, 15 figures, Added a discussion about Holography

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Cited by in corpus (3)

• Exact critical-temperature bounds for two-dimensional Ising models
• Topological states and flat bands in exactly solvable decorated Cayley trees
• Holographic Aspects of Dynamical Mean-Field Theory