Critical Phenomena in Hyperbolic Space

TL;DR

This paper investigates the critical behavior of an $N$-component ${}^{4}$-model in hyperbolic space, revealing a second-order phase transition with unique magnetization textures influenced by the space's curvature.

Contribution

It introduces a new strong curvature fixed point governing phase transitions in hyperbolic space, with unique scaling properties and no order $1/N$ corrections.

Findings

Identifies a second-order phase transition in hyperbolic space.

Discovers a new strong curvature fixed point with specific critical exponents.

Shows no order $1/N$ corrections occur in this setting.

Abstract

In this paper we study the critical behavior of an $N$-component ${\phi}^{4}$-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.