Helical metals and insulators: Sheet singularity of the inflated Berry monopole

We study phases of interacting Dirac matter that host Berry signatures. We predict a topological Lifshitz phase transition caused by the changes of a Dirac cone intersection from a semimetallic phase to helical insulating or metallic phases. These helical phases provide examples of a gapless topological phase where the spectral gap is not required for a topological protection. To realize nodal helical phases one would need to consider isotropic infinite-range interparticle interaction. This interaction could emerge because of a momentum conserving scattering of electrons from a bosonic mode. For repulsive/attractive inter-particle interaction in density/pseudospin channel, the system undergoes a transition to the helical insulator phase. For an attractive density-density interaction, a metallic phase forms that hosts a nodal circle and a nodal sphere in two and three dimensions, respectively. A sheet singularity of Berry curvature is highlighted as a peculiar feature of the nodal sphere phase in three dimensions and represents the extension of the Berry monopole singularities into an inflated monopole. To illustrate the properties of these helical phases we investigate Landau levels in both metallic and insulating phases. Our study provides an extension of the paradigm in the interacting Dirac matter and makes an interesting connection to inflated topological singularities in cosmology.