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Landau levels, Bardeen polynomials and Fermi arcs in Weyl semimetals: the who's who of the chiral anomaly

Condensed matter systems realizing Weyl fermions exhibit striking phenomenology derived from their topologically protected surface states as well as chiral anomalies induced by electromagnetic fields. More recently, inhomogeneous strain or magnetization were predicted to result in chiral electric $\mathbf{E}_5$ and magnetic $\mathbf{B}_5$ fields, which modify and enrich the chiral anomaly with additional terms. In this work, we develop a lattice-based approach to describe the chiral anomaly, which involves Landau and pseudo-Landau levels and treats all anomalous terms on equal footing, while naturally incorporating Fermi arcs. We exemplify its potential by physically interpreting the largely overlooked role of Fermi arcs in the covariant (Fermi level) contribution to the anomaly and revisiting the factor of $1/3$ difference between the covariant and consistent (complete band) contributions to the $\mathbf{E}_5\cdot\mathbf{B}_5$ term in the anomaly. Our framework provides a versatile tool for the analysis of anomalies in realistic lattice models as well as a source of simple physical intuition for understanding strained and magnetized inhomogeneous Weyl semimetals.