  In a quantum Hall system, the finite-wavevector Hall conductivity displays an
intriguing dependence on the Hall viscosity, a coefficient that describes the
non-dissipative response of the fluid to a velocity gradient. In this paper, we
pursue this connection in detail for quantum Hall systems on a lattice, noting
that the neat continuum relation breaks down and develops corrections due to
the broken rotational symmetry. In the process, we introduce a new, quantum
mechanical derivation of the finite-wavevector Hall conductivity for the
integer quantum Hall effect, which allows terms to arbitrary order in the
wavevector expansion to be calculated straightforwardly. We also develop a
universal formalism for studying quantum Hall physics on a lattice, and find
that at weak applied magnetic fields, generic lattice wavefunctions connect
smoothly to the Landau levels of the continuum. At moderate field strengths,
the lattice corrections can be significant and perturb the wavefunctions,
energy levels, and transport properties from their continuum values. Our
approach allows the finite-field behaviour of a system to be inferred directly
from the zero-field band structure.
