What spatial geometry does the (2+1)-dimensional QFT vacuum prefer?

TL;DR

This paper investigates how the vacuum free energy in (2+1)-dimensional quantum field theories influences spatial geometry, revealing a tendency for the vacuum to favor crumpled configurations, with implications for materials like graphene.

Contribution

It demonstrates that quantum vacuum effects in (2+1)-QFTs favor crumpled geometries and are significant for Dirac materials like graphene at room temperature.

Findings

Vacuum free energy difference from flat space is negative for perturbations.

Quantum effects are significant for Dirac fields in graphene at room temperature.

Vacuum energy effects should be considered in modeling graphene's equilibrium shape.

Abstract

We consider relativistic (2+1)-QFTs on a product of time with a two-space and study the vacuum free energy as a functional of the temperature and spatial geometry. We focus on free scalar and Dirac fields on arbitrary perturbations of flat space, finding that the free energy difference from flat space is finite and always \textit{negative} to leading order in the perturbation. Thus free (2+1)-QFTs appear to always energetically favor a crumpled space on all scales; at zero temperature this is a purely quantum effect. Importantly, we show that this quantum effect is non-negligible for the relativistic Dirac degrees of freedom on monolayer graphene even at room temperature, so we argue that this vacuum energy effect should be included for a proper analysis of the equilibrium configuration of graphene or similar materials.