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

We study the new phases of interacting Dirac matter that host novel Berry\nsignatures. We predict a topological Lifshitz phase transition caused by the\nchanges of a Dirac cone intersection from a semimetalic phase to helical\ninsulating or metallic phases. These helical phases provide the examples of\ngapless topological phase where spectral gap is not required for a topological\nprotection. To realize nodal helical phases one would need to consider\nisotropic infinite-range inter-particle interaction. This interaction could\nemerge because of a momentum conserving scattering of electron from a bosonic\nmode. For repulsive/attractive interactions in density/pseudospin channel\nsystem undergoes a transition to helical insulator phase. For an attractive\ndensity-density interaction, a new metallic phase forms that hosts {\\it nodal\ncircle} and {\\it nodal sphere} in two and three dimensions, respectively. A\n{\\it sheet singularity} of Berry curvature is highlighted as a peculiar feature\nof the nodal sphere phase in 3D and represent the extension of the Berry\nmonopole singularities into inflated monopole. To illustrate the properties of\nthese helical phases we investigate Landau levels in both metallic and\ninsulating phases. Our study provides an extension of the paradigm in the\ninteracting Dirac matter and makes an interesting connection to inflated\ntopological singularities in cosmology.\n