Path integral approach to quantum anomalies in interacting models

Abstract

The prediction and subsequent discovery of topological semimetal phases of matter in solid state systems has instigated a surge of activity investigating the exotic properties of these unusual materials. Among these are transport signatures which can be attributed to the chiral anomaly; the breaking of classical chiral symmetry in a quantum theory. This remarkable quantum phenomenon, first discovered in the context of particle physics has now found new life in condensed-matter physics, connecting topological quantum matter and band theory with effective field theoretic models. In this paper we investigate the interplay between interactions and the chiral anomaly in field theories inspired by semimetals using Fujikawa's path integral method. Starting from models in one spatial dimension we discuss how the presence of interactions can affect the consequences of the chiral anomaly leading to renormalization of excitations and their transport properties. This is then generalized to the three-dimensional case where we show that the anomalous response of the system, namely, the chiral magnetic and quantum Hall effects, are modified by the presence of interactions. These properties are investigated further through the identification of anomalous modes which exist within interacting Weyl semimetals. These massive excitations are nonperturbative in nature and are a direct consequence of the chiral anomaly. The effects of interactions on mixed axial-gravitational anomalies are then investigated and the conditions required for interaction effects to be observed are discussed.