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

References in corpus (12)

• Crystallography of Hyperbolic Lattices
• Observation of novel topological states in hyperbolic lattices
• Hyperbolic band theory
• Topological hyperbolic lattices
• Band theory and boundary modes of high-dimensional representations of infinite hyperbolic lattices
• Ising model on hyperbolic lattice studied by corner transfer matrix renormalization group method
• Weak correlation effects in the Ising model on triangular-tiled hyperbolic lattices
• Thermodynamics and structure of simple liquids in the hyperbolic plane
• The Random-Bond Ising Model and its dual in Hyperbolic Spaces
• Dynamic mass generation on two-dimensional electronic hyperbolic lattices
• Statistical Mechanics Approach to the Holographic Renormalization Group: Bethe Lattice Ising Model and p-adic AdS/CFT
• Holographic analysis of boundary correlation functions for the hyperbolic-lattice Ising model

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