 Symmetry-broken magnetisation versus a symmetric ensemble

An even double-well Landau free energy can have stable homogeneous phases at $M=\pm M_0$. In a selected phase, for example the thermodynamic limit with an infinitesimal symmetry-breaking field, mean magnetisation approaches one of those values. An exactly symmetry-preserving finite-system ensemble instead averages both signs and can have mean zero. This ensemble average does not make $M=0$ a stable homogeneous minimum. Distinguish a free-energy minimum, a domain mixture and an unrestricted ensemble average when discussing spontaneous magnetisation.