I

Over the past decade, researchers have linked the low-energy field theory of a Weyl semimetal gapped with a charge-density wave (CDW) to high-energy theories with axion electrodynamics. The possibility of realizing axion-insulating (AXI) phases in Weyl-CDWs has over the past two years returned to the forefront of condensed-matter physics, due to the discovery of a Weyl-semimetal-insulator transition in the established CDW compound (TaSe₄)₂I. In this talk, we will first present theoretical analysis and experimental data establishing that the room-temperature state of (TaSe₄)₂I is indeed a structurally chiral, quasi-1D Weyl semimetal, and that the Weyl-semimetal state becomes gapped by a CDW when cooled just below room temperature. We will next revisit the initial proposals of AXI phases in time-reversal-breaking Weyl-CDWs from the perspective of Magnetic Topological Quantum Chemistry. We will demonstrate that when the low-energy theory of the simplest, inversion-symmetric Weyl-CDW is lattice-regularized in a tight-binding model, the bulk at static values of the CDW phase angle Φ is not an AXI, but in fact realizes one of two quantum anomalous Hall (QAH) phases that differ by a fractional translation in the modulated cell, analogous to the two phases of the Su-Schrieffer-Heeger model of polyacetylene. Crucially, the two QAH phases still differ by an origin-dependent axion angle △θ=π, indicating that the axionic Weyl-CDW response in the first proposals originates from mean-field band topology. We will conclude by considering nonmagnetic Weyl- and Dirac-CDWs, which are non-axionic in the absence of emergent valley symmetry, as θ mod 2π=0 for all static values of Φ. In particular, we will elucidate the relationship between nonmagnetic semimetal-CDWs and non-axionic (helical) higher-order topological crystalline insulators, in which the nontrivial bulk topology emerges from a response coefficient distinct from the axion angle θ.