  We study the new phases of interacting Dirac matter that host novel Berry
signatures. We predict a topological Lifshitz phase transition caused by the
changes of a Dirac cone intersection from a semimetalic phase to helical
insulating or metallic phases. These helical phases provide the examples of
gapless topological phase where spectral gap is not required for a topological
protection. To realize nodal helical phases one would need to consider
isotropic infinite-range inter-particle interaction. This interaction could
emerge because of a momentum conserving scattering of electron from a bosonic
mode. For repulsive/attractive interactions in density/pseudospin channel
system undergoes a transition to helical insulator phase. For an attractive
density-density interaction, a new metallic phase forms that hosts {\it nodal
circle} and {\it nodal sphere} in two and three dimensions, respectively. A
{\it sheet singularity} of Berry curvature is highlighted as a peculiar feature
of the nodal sphere phase in 3D and represent the extension of the Berry
monopole singularities into 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.
