Axion coupling in the hybrid Wannier representation

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 $Z_2$” 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 $Z_2$ 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 $Z_2$ 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.

Figure 1

Illustrative sketches of possible hybrid Wannier (i.e., Wilson loop) band structures for a model with four occupied bands. Hybrid Wannier (HW) centers $z_{nl}\left(\mathbf{k}\right)$ repeat along the vertical direction with period $c$. The horizontal axis represents some path connecting high-symmetry points in the 2D Brillouin zone. (a) Four isolated Wannier bands. (b) Two connected groups of Wannier bands. (c) Fully connected Wannier bands.

Figure 2

(a) An isolated Wannier band (green) lying entirely inside the home unit cell (cutting surfaces, blue, at $z=0$ and $z=c$). (b) The same Wannier band, now represented by two disconnected pieces in the home unit cell, after the cutting surface has been raised. (c) The region ${\mathcal{S}}_n$ where the Wannier band falls below the new cutting surface, and its boundary ${\mathcal{C}}_n$.

Figure 3

Sketch of degeneracy region (shaded) associated with two Wannier bands $z_n\left(\mathbf{k}\right)$ near a point node, and its projection onto the disk-shaped region ${\mathcal{R}}_{\alpha }$ in the 2DBZ (bottom).

Figure 4

Sketches of hybrid Wannier band structures along a path in k-space that captures all the degeneracies of a system with an axion-odd symmetry that reverses $\widehat{\mathbf{z}}$. (a) Disconnected Wannier band structure, decomposed into an origin-centered group (blue shading), a boundary-centered group (red shading), and an uncentered collection (yellow shading). The axion index depends on whether the Wannier bands in the red region have an odd Chern index, which in turn is determined by a counting of Dirac nodes; here $\theta =0$ because the number of nodes in that region is even. (b) Connected Wannier band structure, decomposed into a thin slice centered at $z=c/2$ (red shading) and the remainder (blue shading). Here $\theta =\pi$ since there is an odd number of Dirac cones in the red region.

Figure 5

Wannier band structures for the FKM model in a trivial-insulator phase with $\mathrm{\Delta }t=2.5$ (a), (d), a weak-TI phase with $\mathrm{\Delta }t=-1.0$ (b), (e), and a strong-TI phase with $\mathrm{\Delta }t=1.0$ (c), (f). In (a)–(c) the wannierization is along the (111) direction, denoted as ${\widehat{\mathbf{z}}}^{\prime }$, with lattice periodicity $c^{\prime }=a\sqrt{3}/3$; in (d)–(f) we wannierize along (001), with lattice periodicity $c=a/2$. The bands are plotted along high-symmetry lines connecting the four PTRIM in the 2DBZ.

Figure 6

Wannier band structures for the pyrochlore model, wannierized along (111) (denoted ${\widehat{\mathbf{z}}}^{\prime }$) in (a)–(c) and along (001) in (d)–(f). (a), (d) Trivial insulator; (b), (e) Axion I phase; (c), (f) Axion II phase (see caption of Table 2).

Figure 7

Chern index for connected groups of slab Wannier bands of the pyrochlore model, plotted as a function of their average positions ${\overline{z}}^{\prime }$. Red and blue markers correspond to groups with integer and half-integer ${\overline{z}}^{\prime }/c^{\prime }$, respectively. We consider slabs with a thickness of 10 unit cells along (111) in (a) the trivial phase of Fig. 6, and (b) the axion II phase of Fig. 6; in the axion phase, the top layer was “exfoliated” to remove unwanted nontopological surface states. In the trivial phase, both the groups with integer ${\overline{z}}^{\prime }/c^{\prime }$—one band—and those with half-integer ${\overline{z}}^{\prime }/c^{\prime }$—three bands—have zero Chern index. In the axion II phase both groups comprise two bands each, and far from the surfaces their Chern indices are $-1$ and $+1,$ respectively.

Figure 8

Wannier band structures for the two mirror-symmetric phases of the alternating Haldane model parametrized by the angle $\varphi$ in Eq. (53). (a), (c) is the trivial-insulator phase with $\varphi =0$, and (b), (d) is the axion-insulator phase with $\varphi =\pi$. The wannierization direction is (001)—normal to the mirror plane—in (a), (b), and (010) (denoted ${\widehat{\mathbf{z}}}^{\prime }$)—lying in the mirror plane—in (c), (d). In (a), (b), we indicate the Chern numbers of the Wannier bands to emphasize that while the Wannier band structures look identical, they actually describe different phases.

Figure 9

Chern index of each isolated slab Wannier band of the alternating Haldane model, plotted as a function of its average position $\overline{z}$. We consider slabs with a thickness of 10 unit cells along (001) in (a) the trivial phase of Fig. 8, and (b) the axion phase of Fig. 8. In the trivial phase each Wannier band carries a zero Chern index, while in the axion-odd phase the Chern index alternates between $-1$ and $+1$.

Figure 10

Wannier band structures for the model with glide mirror symmetry. The trivial and axion phases were obtained by setting $(m,\varphi )=(3.5,0.4)$ and $(m,\varphi )=(2.0,0.4),$ respectively, in Eq. (54). In (a), (b) the wannierization direction is $\widehat{\mathbf{x}}\parallel \left(100\right)$, lying in the glide plane and along the half-translation; in (c), (d) it is $\widehat{\mathbf{z}}\parallel \left(001\right)$, normal to the glide plane. The Chern numbers of isolated bands are indicated in panels (a)–(c).