Exotic Thermal Transitions with Spontaneous Symmetry Breaking

We show that exotic thermal transitions with spontaneous symmetry breaking are hosted by thermal phases with topological orders. Our analysis is controlled by finding and perturbing the exact solutions of quantum rotor models coupled to the three-dimensional toric code. We evaluate all critical exponents and find striking characteristics of the exotic transitions. Remarkably, the Ising and XY transitions have the same universality class, and the exotic transitions are more stable than the Wilson-Fisher classes under couplings to acoustic phonons and Fermi surfaces. Thus, topological orders characterize thermal transitions with spontaneous symmetry breaking in sharp contrast to the conventional wisdom stating that order parameter symmetry and spatial dimension sorely determine the universality classes of thermal transitions. Applying our results to experiments in strongly correlated systems, we provide a plausible scenario in puzzlings of doped RbFe$_2$As$_2$.