On the 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. Amongst 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 generalised 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 interactions effects to be observed are discussed.

Non-Hermitian chiral anomalies

“…When the gauge field V is real, W is absent and M = ½ d×d , the above dµ j 5,µ are in agreement with the Hermitian results [25,26]. We note that in the absence of interactions, the above relations reproduce results of Ref.…”

“…While chiral anomaly in condensed matter physics usually describes noninteracting massless fermions under electromagnetic fields, generalizations of this theory allow incorporating Weyl node-mixing terms [23] and treating short-range interactions between fermions [24][25][26]. The latter unveils novel contributions to the anomaly equations through its nonperturbative formulation.…”

“…The current-current interaction with interaction strengths λ 2 µν = λ 2 g µd g νd describes a densitydensity interaction. Similarly, the interaction between chiral currents with λ 2 5,µν = λ 2 5 g µd g νd in d = 4 dimensions embed the spin-spin interaction in its spatial part [26].…”

“…The action S respects U A (1) × U V (1) symmetry classically [25,26], where U A(V ) (1) denotes the chiral (vector) symmetry. However, this classical symmetry does not hold in the presence of quantum fluctuations resulting in the emergence of the chiral anomaly.…”

“…(5) under the vector and chiral transformations, results in satisfying Ã5 a,s = − i dµ j 5,µ a,s and Ãa,s = − i dµ j µ a,s . To obtain the anomalous relations for the interacting model, we should shift the auxiliary fields by their onshell values as a µ → a µ − λ µα j α and s µ → s µ − λ 5µα j 5,α and integrate over Hubbard-Stratonovich fields a and s [26]. The subsequent anomalous equation in the Euclidean space in d = 2 dimensions is…”