Torsional response and Lionville anomaly in Weyl semimetals with dislocations

Weyl nodes in three-dimensional Weyl semimetals break the Liouville equation, leading to the Liouville anomaly. Here we present an approach to derive the semiclassical action and equations of motion for Weyl fermions in the presence of electromagnetic fields and torsions from the quantum field theory: combining the Wigner transformation with band projection operation. It has been shown that the Liouville anomaly, including a pure torsion anomaly term, a mixing term between the electromagnetic fields and torsions, and the conventional chiral anomaly, entirely differs from the counterpart of axial gauge fields. We find various torsional responses and reproduce the chiral vortical effect and the torsional chiral magnetic effect. A torsion-modified anomalous Hall effect due to the mixing term in the Liouville anomaly is predicted, and its implementation is also discussed. Therefore, our work not only provides insights into the torsional responses for Weyl fermions but also acts as a starting point to investigate their topological responses.