Emergence of a Boundary-Sensitive Phase in Hyperbolic Ising Models

TL;DR

This paper uncovers a new boundary-sensitive phase in hyperbolic Ising models, showing that boundary effects induce bulk ordering, and maps out the complete phase diagram including duality relations and boundary-bulk correspondence.

Contribution

It introduces a novel boundary-sensitive intermediate phase in hyperbolic Ising models and develops a specialized numerical method to characterize and confirm its existence and phase transitions.

Findings

Identification of an intermediate phase induced by boundary effects

Numerical evidence supporting three distinct phases and two-stage transitions

Duality relations linking transition points and boundary-bulk correspondence

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.