  We study the propagation of currents along the interface of two $2$-$d$
magnetic systems, where one of them occupies the first quadrant of the plane.
By considering the tight-binding approximation model and K-theory, we prove
that, for an integer number that is given by the difference of two bulk
topological invariants of each individual system, such interface currents are
quantized. We further state the necessary conditions to produce corner states
for these kinds of underlying systems, and we show that they have topologically
protected asymptotic invariants.
