Hall Viscosity and Crystalline Gauge Field in Lattice

The Hall viscosity is a non-dissipative transport coefficient, which becomes non-zero in time-reversal symmetry broken systems such as quantum Hall systems. In the original formulation [Avron et. al., Phys. Rev. Lett. 75, 697 (1995)], the Hall viscosity is expressed in terms of a Berry curvature associated with the change in the metric of the background manifold on which the quantum state lives. Equivalently, the Hall viscosity is given by the Kubo formula involving the stress-energy tensor. However in the lattice setting, both the background metric and the stress-energy tensor may not be well-defined, which results in the main difficulty in properly defining the Hall viscosity in the lattice. In this talk, I will discuss a way to define the Hall viscosity of a lattice model using the idea of twisting the boundary conditions. In particular, I will provide the role of the crystalline gauge field in the Hall viscosity, which in fact plays a role similar to that of the electro-magnetic gauge field in the Hall conductivity.