Occupancy Driven Phase Transitions in Gapped Systems

Systems characterized by an energy gap between the ground state and excited states support a growing number of thermal excitations upon raising the temperature. Provided these excitations interact with each other, their number affects the size of the energy gap akin to a harmonic oscillator where the oscillation frequency is damping dependent. Here we show that once a critical number of excitations is reached, the excitation population becomes thermodynamically unstable and the system transitions between two states. This transition is marked by a divergence in the specific heat, which does not have to be accompanied by an order parameter, nor by the full closure of the gap. We illustrate the thermodynamics using a simple two level system and superfluid ${}^4$He as examples, and we argue that the hidden order transition in URu${}_2$Si${}_2$ is an example of such an occupancy driven transition in the absence of an order parameter.