  Chern insulators arguably provide the simplest examples of topological
phases. They are characterized by a topological invariant and can be identified
by the presence of protected edge states. In this article, we show that a local
impurity in a Chern insulator induces a twofold response: bound states that
carry a chiral current and a net current circulating around the impurity. This
is a manifestation of broken time symmetry and persists even for an
infinitesimal impurity potential. To illustrate this, we consider a Coulomb
impurity in the Haldane model. Working in the low-energy long-wavelength limit,
we show that an infinitesimal impurity strength suffices to create bound
states. We find analytic wavefunctions for the bound states and show that they
carry a circulating current. In contrast, in the case of a trivial analogue,
graphene with a gap induced by a sublattice potential, bound states occur but
carry no current. In the many body problem of the Haldane model at
half-filling, we use a linear response approach to demonstrate a circulating
current around the impurity. Impurity textures in insulators are generally
expected to decay exponentially; in contrast, this current decays polynomially
with distance from the impurity. Going beyond the Haldane model, we consider
the case of coexisting trivial and non-trivial masses. We find that the
impurity induces a local chiral current as long as time reversal symmetry is
broken. However, the decay of this local current bears a signature of the
overall topology - the current decays polynomially in a non-trivial system and
exponentially in a trivial system. In all cases, our analytic results agree
well with numerical tight-binding simulations.
