Second-order geometric responses and central charge*

We extend the study of Hall viscosity and thermal Hall conductivity to spatially inhomogeneous perturbations of the metric; we mainly focus on contributions that are second-order in wavevector.
In a simple non-interacting Quantum Hall model, we confirm[1] a second-order contribution to the antisymmetric part of viscosity area-preserving perturbations (shears), as well as another independent contribution due to dilations.[2] In explicitly calculating geometric linear response coefficients on the edge, we establish a bulk-boundary correspondence for the second-order, antisymmetric part of viscosity.
We substantiate this correspondence using effective field theories and an anomaly inflow argument, which also allows us to relate parts of the second-order geometric responses in the bulk to the topological central charge of the edge theory.

[1] B. Bradlyn, N. Read, Phys. Rev. B 91, 165306 (2015)
[2] A. Abanov, A. Gromov, Phys. Rev. B 90, 014435 (2014)