  The topological response to external perturbations is an effective probe to
characterize different topological phases of matter. Besides the Hall
conductance, the Hall viscosity is another example of such a response that
measures how electronic wave functions respond to changes in the underlying
geometry. Topological (Chern) insulators are known to have a quantized Hall
conductance. A natural question is how the Hall viscosity behaves for these
materials. So far, most of studies on the Hall viscosity of Chern insulators
have focused on the continuum limit. The presence of lattice breaks the
continuous translational symmetry to a discrete group and this causes two
complications: it introduces a new length scale associated with the lattice
constant, and makes the momentum periodic. We develop two different methods of
how to implement a lattice deformation: (1) a lattice distortion is encoded as
a shift in the lattice momentum, and (2) a lattice deformation is treated
microscopically in the gradient expansion of the hopping matrix elements. After
establishing the method of deformation we can compute the Hall viscosity
through a linear response (Kubo) formula. We examine these methods for three
models: the Hofstadter model, the Chern insulator, and the surface of a 3D
topological insulator. Our results in certain regimes of parameters, where the
continuum limit is relevant, are in agreement with previous calculations. We
also provide possible experimental signatures of the Hall viscosity by studying
the phononic properties of a single crystal 3D topological insulator.
