  The absence of an energy gap separating valence and conduction bands makes
the low-energy electronic properties of graphene and its multi-layers sensitive
to electron-electron interactions. In bilayers, for instance, interactions are
predicted to open a gap at charge neutrality, turning the system into an
insulator, as observed experimentally. In mono and (Bernal-stacked) trilayers,
interactions, although still important, do not have an equally drastic effect,
and these systems remain conducting at low temperature. It may be expected that
interaction effects become weaker for thicker multilayers, whose behavior
should eventually converge to that of graphite. Here we show that this
expectation does not correspond to reality by investigating the case of
Bernal-stacked tetralayer graphene (4LG). We reveal the occurrence of a robust
insulating state in a narrow range of carrier densities around charge
neutrality, incompatible with the behavior expected from the single-particle
band structure. The phenomenology resembles that observed in bilayers, but the
stronger conductance suppression makes the insulating state in 4LG visible at
higher temperature. To account for our findings, we suggest a natural
generalization of the interaction-driven, symmetry-broken states proposed for
bilayers. This generalization also explains the systematic even-odd effect of
interactions in Bernal-stacked layers of different thickness that is emerging
from experiments, and has implications for the multilayer-to-graphite
crossover.
