Higher-Order Topology and Defect States in the Charge-Density-Wave Phase of (TaSe$_4$)$_2$I

Recent theoretical and experimental investigations have identified the quasi-1D compound (TaSe$_4$)$_2$I as hosting a Weyl semimetal phase that becomes gapped by an incommensurate charge-density wave (CDW) just below room temperature. Though the CDW phase have been shown to exhibit incipient experimental signatures of (valley-) axion electrodynamics, however, the bulk topology of the insulating CDW state remains an open question. In this talk, we present a physically motivated, lattice-commensurate coupled-wire model based on Topological Quantum Chemistry and crystalline symmetry that approximates the CDW phase of (TaSe$_4$)$_2$I. We demonstrate that our model hosts several higher-order and weak topological phases, depending on the pinned values of several symmetry-allowed mass terms, which we link to independent CDW phase angles $\phi$. We also present evidence for helical modes bound to real-space disclinations and mass vortices in the CDW state.