Nieh-Yan anomaly: Torsional Landau levels, central charge, and anomalous thermal Hall effect

The Nieh-Yan anomaly is the anomalous breakdown of the chiral U(1) symmetry caused by the interaction between torsion and fermions. We study this anomaly from the point of view of torsional Landau levels. It was found that the torsional Landau levels are gapless, while their contributions to the chiral anomaly are canceled, except those from the lowest torsional Landau levels. Hence, the dimension is effectively reduced from $(3+1)$-dimensional to $(1+1)$-dimensional. We further show that the coefficient of the Nieh-Yan anomaly is the free-energy density in $(1+1)$ dimensions. Especially, at finite temperature, the thermal Nieh-Yan anomaly is proportional to the central charge. The anomalous thermal Hall conductance in Weyl semimetals is then shown to be proportional to the central charge, which is the experimental fingerprint of the thermal Nieh-Yan anomaly.