Renormalization flow towards gravitational catalysis in the3dGross-Neveu model

Abstract

Catalyzed symmetry breaking arises from a parametric enhancement of critical fluctuations independently of the coupling strength. Symmetry-breaking fermionic long-range fluctuations exhibit such an enhancement on negatively curved spaces, as is known from mean-field studies. We study gravitational catalysis from the viewpoint of the functional renormalization group using the $3d$ Gross-Neveu model as a specific example. We observe gravitational catalysis toward a phase of broken discrete chiral symmetry both on a maximally symmetric (anti--de Sitter) and on a purely spatially curved manifold for constant negative curvature (Lobachevsky plane). The resulting picture for gravitational catalysis obtained from the renormalization flow is closely related to that of magnetic catalysis. As an application, we estimate the curvature required for subcritical systems of finite length to acquire a gravitationally catalyzed gap.