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

TL;DR

The paper investigates 2+1D CFTs with a conserved U(1) charge in a zero-mean periodic potential, comparing strong-coupling holographic results from Einstein-Maxwell theory in AdS$_4$ to weakly-coupled Dirac-fermion theories. It demonstrates that the IR fixed point flows continuously with the dimensionless ratio $V/k$ in the holographic model, while the weakly-coupled theory exhibits discrete transitions with emergent Dirac points as $V/k$ crosses critical values. Conductivity calculations reveal qualitative similarities between the two approaches (notably a peak near $\omega \sim k$ and a dip for larger frequencies) but also key differences: the discrete IR transitions in the field theory are not captured by the holographic description. The study highlights where holography succeeds in capturing universal, scale-dependent transport and where it requires additional IR degrees of freedom (e.g., monopole operators) to reproduce lattice-induced Fermi-surface physics, with implications for how chemical potentials and lattice effects shape IR CFTs.

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: $μ(x) = V \cos(kx)$. 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.

Figures (22)

• Figure 1: Frequency-dependent conductivity at $V/k = 0.5$ for $N_f$ Dirac fermions in a periodic chemical potential. Here $\sigma_\infty = \sigma (\omega \rightarrow\infty)$ and the Dirac CFT has $\sigma_\infty = N_f/16$.
• Figure 2: A plot of the analytic solution from holography for $\mathrm{Re}(\sigma)$ vs. $\omega$ in the perturbative regime $V \ll k$, normalized to emphasize the strength of the perturbations.
• Figure 3: D.C. conductivity for $N_f$ Dirac fermions in a periodic chemical potential as a function of $V/k$.
• Figure 4: We show the holographic computation of $\sigma(0)$ (approximated by $\sigma(0.01)$, as our numerics cannot compute the d.c. conductivity directly) as a function of $V/k$. This data was taken at $k/T=8$.
• Figure 5: We compare the expectation values of simple operators in the CFT between our first-order perturbative expressions and numerical results. In the numerical results, we have subtracted off the thermal contribution to the stress tensor given by Eq. (\ref{['thermalstress']}). As is clear, we find excellent agreement with the numerics.