Mixed anomalies: Chiral vortical effect and the Sommerfeld expansion

We discuss the connection between the integer moments of the Fermi distribution function that occur in the Sommerfeld expansion and the coefficients that occur in anomalous conservation laws for chiral fermions. For an illustration, we extract the chiral magnetothermal energy current from the mixed gauge-gravity anomaly in the $3+1$-dimensional energy-momentum conservation law. We then use a similar method to confirm the conjecture that the $T^2/12$ thermal contribution to the chiral vortical effect current arises from the gravitational Pontryagin term in the $3+1$-dimensional chiral anomaly.

Figure 1

The Euclidean, imaginary time, section of the two-dimensional black hole is asymptotically a cylinder of circumference $2\pi /\kappa$. The horizon is a single point at $\rho =0$, or equivalently $z=z_H$.