Adjoint ferromagnets

We derive the phase structure and thermodynamics of ferromagnets consisting of elementary magnets carrying the adjoint representation of SU ( N ) and coupled through two-body quadratic interactions. Such systems have a continuous SU ( N ) symmetry as well as a discrete conjugation symmetry. We uncover a rich spectrum of phases and transitions, involving a paramagnetic and two distinct ferromagnetic phases that can coexist as stable and metastable states in different combinations over a range of temperatures. The ferromagnetic phases break SU ( N ) invariance in various channels, leading to spontaneous magnetization. Interestingly, the conjugation symmetry also breaks over a range of temperatures and group ranks N , providing a realization of a spontaneously broken discrete symmetry.