Dirac and Weyl semimetals with holographic interactions

Abstract

Dirac and Weyl semimetals are states of matter exhibiting the relativistic physics of, respectively, the Dirac and Weyl equation in a three-dimensional bulk material. These three-dimensional semimetals have recently been realized experimentally in various crystals. Theoretically, especially the noninteracting case and the case with Coulomb interactions have been studied. However, when a Dirac or Weyl semimetal is coupled to a critical system nearby a quantum critical point, this can induce strong correlations between the fermions, that are not necessarily of Coulombic nature. Perturbation theory fails here, but holographic techniques based on ideas from string theory can provide a theoretical description. We construct a holographic model especially geared to compute the single-particle correlation function of strongly coupled Dirac and Weyl fermions in 3+1 dimensions. Most importantly for our purposes, this correlation function satisfies the zeroth-order frequency sum rule, which makes it a feasible candidate for applications in realistic solid-state materials, e.g., by a direct comparison to angle-resolved photoemission spectroscopy (ARPES) experiments. This so-called semiholographic model for Dirac and Weyl semimetals incorporates the interaction effects corresponding to the critical system via a dual gravitational description. We show how the low-energy behavior of two experimentally measurable quantities, the spectral function and the electrical conductivity, is drastically changed depending on the universality class of the critical point. Results for the spectral function show that there are no long-lived quasiparticles in the relativistic case, due to holographic self-energy effects exhibiting a power-law behavior. The precise power is related to the critical exponent which is a parameter in the semiholographic model. In the case of Lifshitz scaling of the quantum critical point, we find, in contrast to the relativistic case, the existence of a quantum phase transition from a non-Fermi liquid into a Fermi liquid in which two Fermi surfaces spontaneously form, even at zero chemical potential. We also show preliminary results for the case of nonzero chemical potential and Lifshitz scaling. The optical conductivity at zero chemical potential for relativistic scaling consists of two contributions. The interband contribution scales as a power law either in frequency or in temperature for low frequency, and again the precise power is related to the universality class of the quantum critical point. On top of that we find for nonzero temperatures a Drude-like peak corresponding to intraband transitions. The dc conductivity is finite in the presence of translational invariance, on account of particle-hole symmetry in combination with the interactions which cause the charge current to relax. A behavior similar to Coulomb interactions is recovered as a special limiting case. Finally, we present two possible field-theoretic derivations of these results, using either a semiholographic or a holographic point of view. In the semiholographic interpretation, we also show how, in general, the conductivity should be calculated in agreement with Ward identities. The resulting field-theory interpretation may lead to a better understanding of the holographic dictionary in applied holography.