Lorentz violation in Dirac and Weyl semimetals

We propose a correspondence between the description of emergent Lorentz symmetry in condensed-matter systems and the established general effective field theory for Lorentz violation in fundamental theories of spacetime and matter. This correspondence has potential implications in both directions. We illustrate the proposal by investigating its consequences for the spectral and transport properties of Dirac and Weyl semimetals. Particular realizations of this framework give rise to Dirac nodal spectra with nodal lines and rings. We demonstrate a bulk-boundary correspondence between bulk topological invariants and drumhead surface states of these Dirac nodal semimetals. We calculate their transport coefficients in leading-order perturbation theory, thereby characterizing the unconventional electromagnetic response due to small deviations from emergent Lorentz invariance. Some prospective future applications of the correspondence are outlined.