  Spontaneous symmetry breaking (SSB) in statistical physics is a macroscopic
collective phenomenon. For the paradigmatic Q-state Potts model it means a
transition from the disordered color-symmetric phase to an ordered phase in
which one color dominates. Existing mean field theories imply that SSB in the
microcanonical statistical ensemble (with energy being the control parameter)
should be a continuous process. Here we study microcanonical SSB on the
random-graph Potts model, and discover that the entropy is a kinked function of
energy. This kink leads to a discontinuous phase transition at certain energy
density value, characterized by a jump in the density of the dominant color and
a jump in the microcanonical temperature. This discontinuous SSB in random
graphs is confirmed by microcanonical Monte Carlo simulations, and it is also
observed in bond-diluted finite-size lattice systems.
