Surface Luttinger surfaces and Luttinger-Lifshitz transitions in topological band structures

The standard paradigm of topological phases posits that the surface two phases with the same symmetries are necessarily separated by a bulk phase transition, while breaking the symmetry unlocks a path in parameter space that allows the phases to be connected adiabatically. Moreover, if the symmetry is broken only on the boundary, single-particle properties are expected to be blind to distinction between the two phases. In this work, we prove that this expectation is incorrect. We first reveal that the single-particle surface Green's function contains zeros or ``LSs'' that respect the same bulk-boundary correspondence that the well-known topological surface states do. At non-zero temperatures, these modes contribute negatively to an effective surface specific heat. Remarkably, the LSs survive symmetry-breaking perturbations that destroy the surface states. Thus, a bulk topological phase transition causes a reconstruction of Luttinger surfaces on the surface and manifests as a discontinuity in the surface specific heat.