Discontinuous Spontaneous-Symmetry-Breaking in the Microcanonical Ensemble

Spontaneous-symmetry-breaking (SSB) is a macroscopic collective phenomenon occurring at phase transitions. Although deep understanding of SSB in the canonical statistical ensemble has been achieved, SSB in the microcanonical ensemble (with energy being the control parameter) has rarely been discussed. Here we study microcanonical SSB on the paradigmatic $Q$-state Potts model ($Q\! \geq\! 3$). The Bethe-Peierls mean field theory predicts a discontinuous phase transition in regular random graphs as the result of entropy competition at fixed energy between the disordered symmetric phase and the microcanonical polarized phase, with a jump in the density of the dominant color and a jump in the microcanonical temperature. This discontinuous SSB is confirmed by microcanonical Monte Carlo simulations, and our simulation results further suggest that it will also occur in finite-dimensional bond-diluted lattice systems.