Robust boundary Luttinger surfaces in topological band structures

The standard paradigm of topological phases posits that two phases with identical symmetries are separated by a bulk phase transition, while symmetry breaking provides a path in parameter space that allows adiabatic connection between the phases. Typically, if symmetry is broken only at the boundary, topological surface states become gapped, and single-particle surface properties no longer distinguish between the two phases. In this work, we challenge this expectation. We demonstrate that the single-particle surface Green's function contains zeros, or "Luttinger surfaces," which maintain the same bulk-boundary correspondence as topological surface states. Remarkably, these Luttinger surfaces persist under symmetry-breaking perturbations that destroy the surface states. Moreover, we point out that low-energy and surface theories, often used synonymously in discussions of (gapped) topological matter, are actually different, with the difference captured by the Luttinger surfaces.