Conformal field theories in a periodic potential: results from holography and field theory

TL;DR

This paper investigates 2+1D conformal field theories with a periodic chemical potential, comparing holographic and weakly-coupled approaches, revealing qualitative similarities and key differences at specific parameter values.

Contribution

It provides a comparative analysis of strongly-coupled holographic and weakly-coupled field theory results for periodically modulated CFTs, highlighting the qualitative similarities and the nature of phase transitions.

Findings

Qualitative similarities in conductivity results between holography and weak coupling.

Discrepancies at specific V/k values where the infrared CFT changes.

Emergence of zero modes and local Fermi surfaces in weakly-coupled theories.

Abstract

We study 2+1 dimensional conformal field theories (CFTs) with a globally conserved U(1) charge, placed in a chemical potential which is periodically modulated along the spatial direction $x$ with zero average: $\mu(x) = V \cos(kx)$. The dynamics of such theories depends only on the dimensionless ratio $V/k$, and we expect that they flow in the infrared to new CFTs whose universality class changes as a function of $V/k$. We compute the frequency-dependent conductivity of strongly-coupled CFTs using holography of the Einstein-Maxwell theory in 4-dimensional anti-de Sitter space. We compare the results with the corresponding computation of weakly-coupled CFTs, perturbed away from the CFT of free, massless Dirac fermions (which describes graphene at low energies). We find that the results of the two computations have significant qualitative similarities. However, differences do appear in the vicinities of an infinite discrete set of values of $V/k$: the universality class of the infrared CFT changes at these values in the weakly-coupled theory, by the emergence of new zero modes of Dirac fermions which are remnants of local Fermi surfaces. The infrared theory changes continuously in holography, and the classical gravitational theory does not capture the physics of the discrete transition points between the infrared CFTs. We briefly note implications for a non-zero average chemical potential.