Hall Viscosity I: Linear Response Theory for Viscosity

In two dimensional systems with broken time-reversal symmetry, there can exist a non-dissipative viscosity coefficient [1,2,3]. This Hall viscosity is similar in nature to the non-dissipative Hall conductivity. In order to investigate this phenomenon further, we develop a linear response formalism for viscosity. We derive a Kubo formula for the frequency dependent viscosity tensor in the long wavelength limit. We compute the viscosity tensor for the free electron gas, integer quantum Hall systems, and two-dimensional paired superfluids. In the zero frequency limit, we show how the known results [3,4] for the Hall viscosity are recovered.\\[4pt] [1] J. Avron, R. Seiler, and P. Zograf, Phys. Rev. Lett. {\bf 75}, 697 (1995).\\[0pt] [2] P. Levay, J. Math. Phys. {\bf 36}, 2792 (1995).\\[0pt] [3] N. Read, Phys. Rev. B {\bf 79}, 045308 (2009).\\[0pt] [4] N. Read and E. Rezayi, Phys. Rev. B {\bf 84}, 085316 (2011).