Critical behavior in spherical and hyperbolic spaces

Abstract

We study the effects of curved background geometries on the critical behavior of scalar field theory. In particular we concentrate on two maximally symmetric spaces: $d$-dimensional spheres and hyperboloids. In the first part of the paper, by applying the Ginzburg criterion, we find that for large correlation length the Gaussian approximation is valid on the hyperboloid for any dimension $d\ge 2$, while it is not trustable on the sphere for any dimension. This is understood in terms of various notions of effective dimension, such as the spectral and Hausdorff dimension. In the second part of the paper, we apply functional renormalization group methods to develop a different perspective on such phenomena, and to deduce them from a renormalization group analysis. By making use of the local potential approximation, we discuss the consequences of having a fixed scale in the renormalization group equations. In particular, we show that in the case of spheres there is no true phase transition, as symmetry restoration always occurs at large scales. In the case of hyperboloids, the phase transition is still present, but as the only true fixed point is the Gaussian one, mean field exponents are valid also in dimensions lower than four.

24 pages, 5 figures; several improvements in the presentation and small corrections, 2 figures added

References in corpus (11)

• Breakdown of Semiclassical Methods in de Sitter Space
• The local potential approximation in quantum gravity
• Tuning the fragility of a glassforming liquid by curving space
• High-accuracy scaling exponents in the local potential approximation
• Scaling Solutions in Continuous Dimension
• Percolation on hyperbolic lattices
• Curvature-induced frustration in the $XY$ model on hyperbolic surfaces
• Triviality from the Exact Renormalization Group
• Novel scaling behavior of the Ising model on curved surfaces
• Wilsonian Renormalization of Noncommutative Scalar Field Theory
• Thermodynamics and structure of simple liquids in the hyperbolic plane