Equivalence between the Axion Invariant and theS4Symmetry Indicator

The equivalence between the axion invariant and thesymmetry indicator is established for three-dimensional-symmetric axion insulators with vanishing three-dimensional Chern numbers. Starting from the Chern-Simons expression for the magnetoelectric polarizability,is rewritten in terms of thesewing matrix. After stable reduction to determinant-one two-band blocks, the invariant is expressed as the degree of a map from the Brillouin zone to. The degree modulo two is then evaluated from theeigenvalues at the four-invariant momenta and is shown to coincide with the symmetry indicator. A minimal tight-binding model withsymmetry verifies the correspondence betweenandthroughout its phase diagram. The result closes a gap between the topological-field-theory description of the axion response and the topological-band-theory classification by symmetry indicators. It also extends the known response-indicator equivalence from antiunitary settings such assymmetry to the unitary, orientation-reversing rotoinversion symmetry.