  Electrons in condensed matter may transition into a variety of
broken-symmetry phase states due to electron-electron interactions. Applying
diverse mean-field approximations to the interaction term is arguably the
simplest way to identify the phase states theoretically possible in a given
setting. Here, we explore electron-electron attraction in a two-band system
comprising symmetric conduction and valence bands touching each other at a
single point. We assume a mean-field pairing between the electrons having
opposite spins, momenta, and, in contrast to the conventional superconducting
pairing, residing in opposite bands, i.e., having opposite energies. We show
that electrons transition into a novel correlated ground state if and only if
the bands are flat enough, i.e. the transition is impossible in the case of
conventional parabolic bands. Although this state is not superconducting in the
usual sense and does not exhibit a gap in its excitation spectrum, it is
nevertheless immune to elastic scattering caused by any kind of disorder, and
is therefore expected to exhibit high electric conductivity at low temperature,
mimicking the behavior of a real superconductor. Having in mind the recent
experimental realizations of flat-band electronic systems in twisted
multilayers, we foresee an exciting opportunity for observing a new class of
highly conductive materials.
