  Suspended Bernal-stacked graphene multilayers up to an unexpectedly large
thickness exhibit a broken-symmetry ground state, whose origin remains to be
understood. Here we show that a finite-temperature second order phase
transition occurs in multilayers whose critical temperature Tc increases from
12 K in bilayers to 100 K in heptalayers. A comparison of the data to a
phenomenological model inspired by a mean field approach suggests that the
transition is associated with the appearance of a self-consistent valley- and
spin-dependent staggered potential changing sign from one layer to the next,
appearing at Tc and increasing upon cooling. The systematic evolution with
thickness of several measured quantities imposes constraints on any microscopic
theory aiming to analyze the nature of electronic correlations in this system.
