Critical Phenomena in Hyperbolic Space

Abstract

In this paper we study the critical behavior of an $N$-component ${\varphi }^4$-model in hyperbolic space, which serves as a model of uniform frustration. We find that this model exhibits a second-order phase transition with an unusual magnetization texture that results from the lack of global parallelism in hyperbolic space. Angular defects occur on length scales comparable to the radius of curvature. This phase transition is governed by a new strong curvature fixed point that obeys scaling below the upper critical dimension $d_{uc}=4$. The exponents of this fixed point are given by the leading order terms of the $1/N$ expansion. In distinction to flat space no order $1/N$ corrections occur. We conclude that the description of many-particle systems in hyperbolic space is a promising avenue to investigate uniform frustration and non-trivial critical behavior within one theoretical approach.

12 pages, 5 figures

References in corpus (9)

• Cavity method for quantum spin glasses on the Bethe lattice
• Tuning the fragility of a glassforming liquid by curving space
• Hopping on the Bethe lattice: Exact results for densities of states and dynamical mean-field theory
• Critical behavior in spherical and hyperbolic spaces
• Ising model on hyperbolic lattice studied by corner transfer matrix renormalization group method
• Curvature-induced frustration in the $XY$ model on hyperbolic surfaces
• Weak correlation effects in the Ising model on triangular-tiled hyperbolic lattices
• Tricritical point of J1-J2 Ising model on hyperbolic lattice
• Ising model on a hyperbolic plane with a boundary

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• Emergence of a Boundary-Sensitive Phase in Hyperbolic Ising Models
• Classical spin liquids from frustrated Ising models in hyperbolic space