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We study the new phases of interacting Dirac matter that host novel Berrysignatures. We predict a topological Lifshitz phase transition caused by thechanges of a Dirac cone intersection from a semimetalic phase to helicalinsulating or metallic phases. These helical phases provide the examples ofgapless topological phase where spectral gap is not required for a topologicalprotection. To realize nodal helical phases one would need to considerisotropic infinite-range inter-particle interaction. This interaction couldemerge because of a momentum conserving scattering of electron from a bosonicmode. For repulsive/attractive interactions in density/pseudospin channelsystem undergoes a transition to helical insulator phase. For an attractivedensity-density interaction, a new metallic phase forms that hosts {\it nodalcircle} and {\it nodal sphere} in two and three dimensions, respectively. A{\it sheet singularity} of Berry curvature is highlighted as a peculiar featureof the nodal sphere phase in 3D and represent the extension of the Berrymonopole singularities into inflated monopole. To illustrate the properties ofthese helical phases we investigate Landau levels in both metallic andinsulating phases. Our study provides an extension of the paradigm in theinteracting Dirac matter and makes an interesting connection to inflatedtopological singularities in cosmology.

Helical metals and insulators and sheet singularity of inflated Berry monopole