  We consider topologically non-trivial interface Hamiltonians, which find
several applications in materials science and geophysical fluid flows. The
non-trivial topology manifests itself in the existence of topologically
protected, asymmetric edge states at the interface between two two-dimensional
half spaces in different topological phases. It is characterized by a quantized
interface conductivity. The objective of this paper is to compute such a
conductivity and show its stability under perturbations. We present two
methods.
  The first one computes the conductivity using the winding number of branches
of absolutely continuous spectrum of the interface Hamiltonian. This
calculation is independent of any bulk properties but requires a sufficient
understanding of the spectral decomposition of the Hamiltonian. In the fluid
flow setting, it also applies in cases where the so-called bulk-interface
correspondence fails.
  The second method establishes a bulk-interface correspondence between the
interface conductivity and a so-called bulk-difference invariant. We introduce
the bulk-difference invariants characterizing pairs of half spaces. We then
relate the interface conductivity to the bulk-difference invariant by means of
a Fedosov-H\"ormander formula, which computes the index of a related Fredholm
operator.
  The two methods are used to compute invariants for representative $2\times2$
and $3\times3$ systems of equations that appear in the applications.
