Thermodynamics of a higher-order topological insulator

We investigate the order of the topological quantum phase transition in a two-dimensional quadrupolar topological insulator within a thermodynamic approach. Using numerical methods, we separate the bulk, edge, and corner contributions to the grand potential and detect different phase transitions in the topological phase diagram. The transitions from the quadrupolar to the trivial or to the dipolar phase are well captured by the thermodynamic potential. On the other hand, we have to resort to a grand potential based on the Wannier bands to describe the transition from the trivial to the dipolar phase. The critical exponents and the order of the phase transitions are determined and discussed in the light of the Josephson hyperscaling relation.