Abstract
The exploration of strongly-interacting finite-density states of matter has been a major recent application of gauge-gravity duality. When the theories involved have a known Lagrangian description, they are typically deformations of large $N$ supersymmetric gauge theories, which are unusual from a condensed-matter point of view. In order to better interpret the strong-coupling results from holography, an understanding of the weak-coupling behavior of such gauge theories would be useful for comparison. We take a first step in this direction by studying several simple supersymmetric and non-supersymmetric toy model gauge theories at zero temperature. Our supersymmetric examples are $\mathcal{N}=1$ super-QED and $\mathcal{N}=2$ super-QED, with finite densities of electron number and R-charge respectively. Despite the fact that fermionic fields couple to the chemical potentials we introduce, the structure of the interaction terms is such that in both of the supersymmetric cases the fermions do not develop a Fermi surface. One might suspect that all of the charge in such theories would be stored in the scalar condensates, but we show that this is not necessarily the case by giving an example of a theory without a Fermi surface where the fermions still manage to contribute to the charge density.
Highlights
The ability to do controlled calculations on the gravity side of the duality comes with several conditions and costs
One might suspect that all of the charge in such theories would be stored in the scalar condensates, but we show that this is not necessarily the case by giving an example of a theory without a Fermi surface where the fermions still manage to contribute to the charge density
We should not expect the behavior of the fermions to be close to that of a conventional free system once there is scalar condensation, because of the structure of the Yukawa interactions and the scalar self-interactions dictated by supersymmetry
Summary
The standard example of a finite-density relativistic system involving fermions and gauge fields is a QED plasma, which we briefly describe before considering supersymmetric theories. Screening effects in an electron plasma are such that the residual Coulomb interaction obliterates the would-be gapless Fermi zero-sound mode present in Fermi liquids, turning it into the gapped plasmon mode of the dense electron gas as explained in e.g. chapter 16 of the textbook [51] Given these results for non-supersymmetric gauge theories at finite density, we clearly cannot assume that the N = 1 and N = 2 sQED plasmas should be Fermi liquids. The mixing alters the dispersion relations of the fermion fields at the quadratic level, and so we cannot assume that the response of the electrons to a chemical potential at g = 0, which involves the formation of a Fermi surface, will necessarily persist to any g > 0, no matter how small This observation is generic, and applies to essentially any supersymmetric gauge theory in which one turns on a chemical potential for selectrons or squarks which can cause selectron or squark condensation
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