In 1982 Nieh and Yan introduced the quantum gravitational anomaly caused by the gravitational torsion field [1, 2]. Since that time the torsional anomaly has been debated, because the coefficient in the Nieh-Yan anomaly term contains the ultraviolet energy cut-off, which is not well defined.

In this paper we discuss the temperature correction to the Nieh-Yan anomaly. As distinct from the zero temperature term, the $T^2$ temperature correction  does not depend on the ultraviolet cut-off and thus can be universal. Such $T^2$ Nieh-Yan term may exist not only in the relativistic quantum field theories, but also in condensed matter with Weyl fermions. In the topological  Weyl semimetals and in the chiral $p+ip$ superfluids and superconductors, this term is fully determined by the quasirelativistic physics in the vicinity of the Weyl nodes.

[1] H. T. Nieh and M. L. Yan, J. Math. Phys. 23, 373  (1982).

[2] H. T. Nieh and M. L. Yan, Ann. Phys.138, 237 (1982).

Nissinen J., Volovik G.E.

JETP Letters 110, issue 12 (2019)