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arXiv:2507.21044 · Physical review. B./Physical review. B

Emergence of a boundary-sensitive phase in hyperbolic Ising models

Xingzhi Wang, Zohar Nussinov, Gerardo Ortíz

Published Oct 21, 2025• 5 citations
arXivHTMLPDF

Abstract

The introduction of spatial curvature leads to novel phases lying beyond the conventional Landau and topological order paradigms. In Ising models on hyperbolic lattices, the authors identify a boundary-sensitive bulk ordered phase that emerges between the ferromagnetic and paramagnetic regimes as the temperature increases. Using a symmetry-restricted corner transfer matrix renormalization group approach, they map here the complete phase diagram with transitions linked by a boundary-transforming Kramers--Wannier duality. Cayley tree limit calculations expose a sharp holographic correspondence between boundary and bulk correlation functions.

DOI
10.1103/f4sj-rwvj

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The introduction of spatial curvature leads to novel phases lying beyond the conventional Landau and topological order paradigms. In Ising models on hyperbolic lattices, the authors identify a boundary-sensitive bulk ordered phase that emerges between the ferromagnetic and paramagnetic regimes as the temperature increases. Using a symmetry-restricted corner transfer matrix renormalization group approach, they map here the complete phase diagram with transitions linked by a boundary-transforming Kramers--Wannier duality. Cayley tree limit calculations expose a sharp holographic correspondence between boundary and bulk correlation functions.
We analyze the thermodynamic properties of the random-bond Ising model (RBIM) on closed hyperbolic surfaces using Monte Carlo and high-temperature series expansion techniques. We also analyze the dual-RBIM, that is, the model that in the absence of disorder is related to the RBIM via the Kramers-Wannier duality. Even on self-dual lattices this model is different from the RBIM, unlike in the Euclidean case. We explain this anomaly by a careful rederivation of the Kramers-Wannier duality. For the (dual-)RBIM, we compute the paramagnet-to-ferromagnet phase transition as a function of both temperature T and the fraction of antiferromagnetic bonds p. We find that as temperature is decreased in the RBIM, the paramagnet gives way to either a ferromagnet or a spin-glass phase via a second-order transition compatible with mean-field behavior. In contrast, the dual-RBIM undergoes a strongly first-order transition from the paramagnet to the ferromagnet both in the absence of disorder and along the Nishimori line. We study both transitions for a variety of hyperbolic tessellations and comment on the role of coordination number and curvature. The extent of the ferromagnetic phase in the dual-RBIM corresponds to the correctable phase of hyperbolic surface codes under independent bit- and phase-flip noise.
Abstract We analyze boundary spin correlation functions of the hyperbolic-lattice Ising model from the holographic point of view. Using the corner-transfer-matrix renormalization-group (CTMRG) method, we demonstrate that the boundary correlation function exhibits power-law decay with quasiperiodic oscillation, while the bulk correlation function always decays exponentially. On the basis of the geometric relation between the bulk correlation path and distance along the outer edge boundary, we find that scaling dimensions for the boundary correlation function can be well explained by a combination of the bulk correlation length and background curvatures inherent to the hyperbolic lattice. We also investigate the cutoff effect of the bond dimension in CTMRG, revealing that the long-distance behavior of the boundary spin correlation is accurately described even with a small bond dimension. In contrast, the short-distance behavior rapidly loses its accuracy.
We show that quantum systems on non-amenable lattices, whose boundaries constitute an extensive fraction of the system, host novel forms of bulk ordering with no Euclidean counterpart. Combining boundary-retaining iDMRG with an exact symmetry-fixing duality, we investigate p<5p<5p<5 state clock models on the coordination-3 infinite Cayley tree. We find that between the ordered and disordered phases lies an emergent intermediate phase with bulk order induced by an arbitrarily weak boundary field rather than by conventional spontaneous symmetry-breaking. This establishes boundary-sensitive order as a distinct quantum ordering mechanism defying Landau's spontaneous symmetry-breaking paradigm. The intermediate phase realizes a bulk-boundary correspondence in which the unpolarized boundary selects a specific ordered bulk state.
Two-dimensional ferromagnetic N -state clock models are studied on a hyperbolic lattice represented by tessellation of pentagons. The lattice lies on the hyperbolic plane with a constant negative scalar curvature. We observe the spontaneous magnetization, the internal energy, and the specific heat at the center of sufficiently large systems, where fixed boundary conditions are imposed, for the cases N>or=3 up to N=30 . The model with N=3 , which is equivalent to the three-state Potts model on the hyperbolic lattice, exhibits a first-order phase transition. A mean-field-like phase transition of second order is observed for the cases N>or=4 . When N>or=5 we observe a Schottky-type specific heat below the transition temperature, where its peak height at low temperatures scales as N(-2) . From these facts we conclude that the phase transition of the classical XY model deep inside hyperbolic lattices is not of the Berezinskii-Kosterlitz-Thouless type.
The matrix product structure is considered on a regular lattice in the hyperbolic plane. The phase transition of the Ising model is observed on the hyperbolic (5,4)(5, 4)(5,4) lattice by means of the corner-transfer-matrix renormalization group (CTMRG) method. Calculated correlation length is always finite even at the transition temperature, where mean-field like behavior is observed. The entanglement entropy is also always finite.
The ferromagnetic-paramagnetic phase transition of the two-dimensional frustrated Ising model on a hyperbolic lattice is investigated by use of the corner transfer matrix renormalization group method. The model contains a ferromagnetic nearest-neighbor interaction J1 and a competing antiferromagnetic interaction J2 . A mean-field-like second-order phase transition is observed when the ratio kappa=J_{2}J_{1} is less than 0.203. In the region 0.203<kappa<14 , the spontaneous magnetization is discontinuous at the transition temperature. Such tricritical behavior suggests that the phase transitions on hyperbolic lattices need not always be mean-field-like.