  The standard paradigm of topological phases posits that 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, topological surface states are generically gapped and single-particle
surface properties are expected to be blind to distinction between the two
phases. In this work, we prove this last expectation incorrect. We first reveal
that the single-particle surface Green's function contains zeros or ``Luttinger
surfaces'' that respect the same bulk-boundary correspondence as the well-known
topological surface states. Remarkably, the Luttinger surfaces survive
symmetry-breaking perturbations that destroy the surface states. Thus, a bulk
topological phase transition in the presence of surface symmetry breaking
causes a reconstruction of Luttinger surfaces on the surface, which we refer to
as a surface Luttinger-Lifshitz transition. At non-zero temperatures, the
Luttinger surfaces contribute negatively to an effective surface specific heat,
and a Luttinger-Lifshitz transition manifests as a discontinuity in this
specific heat.
