Axion coupling in the hybrid Wannier representation

Abstract

Many magnetic point-group symmetries induce a topological classification on crystalline insulators, dividing them into those that have a nonzero quantized Chern-Simons magnetoelectric coupling ("axion-odd" or "topological"), and those that do not ("axion-even" or "trivial"). For time-reversal or inversion symmetries, the resulting topological state is usually denoted as a "strong topological insulator" or an "axion insulator" respectively, but many other symmetries can also protect this "axion Z2" index. Topological states are often insightfully characterized by choosing one crystallographic direction of interest, and inspecting the hybrid Wannier (or equivalently, the non-Abelian Wilson-loop) band structure, considered as a function of the two-dimensional Brillouin zone in the orthogonal directions. Here, we systematically classify the axion-quantizing symmetries, and explore the implications of such symmetries on the Wannier band structure. Conversely, we clarify the conditions under which the axion Z2 index can be deduced from the Wannier band structure. In particular, we identify cases in which a counting of Dirac touchings between Wannier bands, or a calculation of the Chern number of certain Wannier bands, or both, allows for a direct determination of the axion Z2 index. We also discuss when such symmetries impose a "flow" on the Wannier bands, such that they are always glued to higher and lower bands by degeneracies somewhere in the projected Brillouin zone, and the related question of when the corresponding surfaces can remain gapped, thus exhibiting a half-quantized surface anomalous Hall conductivity. Our formal arguments are confirmed and illustrated in the context of tight-binding models for several paradigmatic axion-odd symmetries including time reversal, inversion, simple mirror, and glide mirror symmetries.

Axionic band topology in inversion-symmetric Weyl-charge-density waves

“…In 3D TIs and AXIs, the combination of θ = π and ν x,y,z = 0 leads to unusual response properties, including low-energy excitations resembling magnetic monopoles (the Witten effect [40,41]) and quantized Faraday and Kerr rotations [4,42]. AXIs have recently been recognized as "higher-order" TIs (HOTIs) [28][29][30][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60] featuring gapped surfaces and odd numbers of sample-encircling chiral hinge modes [ Fig. 1(a)].…”

“…1(b) and 1(c)]-that differ by a fractional lattice translation. Crucially, although θ is origin dependent in the presence of a background QAH [4,30,54], we find that the QAH and oQAH phases still differ by an origin-(gauge-) independent, topological axion angle δθ φ = π that reflects a difference in I-quantized "Chern number polarization" [54,81]. This provides a direct analogy between axionic CDWs and the Su-Schrieffer-Heeger model of an I-symmetric CDW in polyacetylene [82], in which both phases are trivial atomic limits that differ by a fractional lattice translation corresponding to an I-quantized topological polarization.…”

“…However, because weak Chern numbers are only I-symmetry-indicated modulo 2 [93,94], |δz 4 | = 2 does not itself indicate an AXI transition. Additionally, when ν z = 0, defining θ uniquely requires the specification of a reference state and I center (i.e., an origin) [54]. Furthermore, if there are other bulk or surface contributions to the topological response (e.g., other massive Dirac fermions at larger momenta, or a background QAH), then defects in V φ will host additional states that coexist with and obscure the AXI bound states.…”

“…To explicitly determine the bulk topology, we will employ model-agnostic numerical methods [28,30,54] to extract ν z (φ) and δθ φ . To begin, we fix the origin to the I center at (x, y, z) = (0, 0, 0) in the modulated cell and form an R idirected, I-symmetric slab.…”

“…To begin, we fix the origin to the I center at (x, y, z) = (0, 0, 0) in the modulated cell and form an R idirected, I-symmetric slab. The Hall conductance G H,i of the slab consists of an extensive contribution from the bulk QAH and an intensive contribution from θ that reflects either the bulk magnetoelectric polarizability or a QAH effect offset from the origin [4,30,54]:…”