We present experimental results for the density of states (DOS) of a superconducting microwave Dirac billiard which serves as an idealized model for the electronic properties of graphene. The DOS exhibits two sharp peaks which evolve into van Hove singularities with increasing system size. They divide the band structure into regions governed by the relativistic Dirac equation and by the nonrelativistic Schrödinger equation, respectively. We demonstrate that in the thermodynamic limit, a topological transition appears as a neck-disrupting Lifshitz transition in the number susceptibility and as an excited-state transition in the electronic excitations. Furthermore, we recover the finite-size scaling typical for excited-state quantum phase transitions involving logarithmic divergences and identify a quasiorder parameter.

• Received 16 April 2013

DOI:https://doi.org/10.1103/PhysRevB.88.104101