Mirror Chern numbers in the hybrid Wannier representation

The topology of electronic states in band insulators with mirror symmetry can be classified in two different ways. One is in terms of the mirror Chern number, an integer that counts the number of protected Dirac cones in the Brillouin zone of high-symmetry surfaces. The other is via a ${\mathbb{Z}}_2$ index that distinguishes between systems that have a nonzero quantized orbital magnetoelectric coupling (“axion-odd”), and those that do not (“axion-even”); this classification can also be induced by other symmetries in the magnetic point group, including time reversal and inversion. A systematic characterization of the axion ${\mathbb{Z}}_2$ topology has previously been obtained by representing the valence states in terms of hybrid Wannier functions localized along one chosen crystallographic direction, and inspecting the associated Wannier band structure. Here we focus on mirror symmetry, and extend that characterization to the mirror Chern number. We choose the direction orthogonal to the mirror plane as the Wannierization direction and show that the mirror Chern number can be determined from the winding numbers of the touching points between Wannier bands on mirror-invariant planes and from the Chern numbers of flat bands pinned to those planes. In this representation, the relation between the mirror Chern number and the axion ${\mathbb{Z}}_2$ index is readily established. The formalism is illustrated by means of ab initio calculations for SnTe in the monolayer and bulk forms, complemented by tight-binding calculations for a toy model.

Figure 1

The upper panel shows schematically a pair of 2D crystals lying on the $(x,z$) plane; each has one atom per primitive cell (black dots), and lattice constant $c$ along $z$. The crystal on the left has a rectangular lattice and a type-1 horizontal mirror, with inequivalent mirror lines $z=0\text{mod}c$ (A) and $z=c/2\text{mod}c$ (B), shown as dashed lines; the one on the right has a centered rectangular lattice and a type-2 mirror, with equivalent mirror lines A and B. The lattice vectors ${\mathbf{a}}_3$ and ${\tilde{\mathbf{a}}}_3$ are defined in the main text. The lower panel shows the reciprocal lattices, with a separation of $2\pi /c$ between horizontal lattice lines G. On the left the periodicity along $k_z$ is $2\pi /c$, and hence both $k_z=0\text{mod}2\pi /c$ (G) and $k_z=\pi /c\text{mod}2\pi /c$ (X) are pointwise-invariant mirror lines, as indicated by the dashed lines. On the right, where the periodicity along $k_z$ is $4\pi /c$, G is a mirror-invariant line but X is not. The associated Brillouin zones are indicated by the shaded green areas.

Figure 2

(a) Atomic structure of monolayer SnTe. The black square is the conventional unit cell with lattice constant $a$, and the red square is the primitive cell with lattice constant $a^{\prime }=a/\sqrt{2}$. (b) Brillouin zone and high-symmetry points.

Figure 3

(a) Energy bands of monolayer SnTe, with the $s$-type lower valence bands that are excluded from the Wannierization shown in gray. All bands are doubly degenerate, and the Fermi level is indicated by the dashed line. (b) Wannier bands obtained from the Bloch states in the six $p$-type upper valence bands. (c) Heat map plot of the gap function of Eq. (52) for the central pair of Wannier bands, where zero-gap points (nodal points) appear as dark spots. Those with winding numbers $W_j=\pm 1$ are indicated by red or blue circles, while the one with $W_j=-3$ at the $\mathrm{\Gamma }$ point is indicated by a blue triangle. Dashed circles denote pairs of nearby nodes with equal and opposite winding numbers. When a node falls on the BZ boundary, only one of the periodic images is shown.

Figure 4

(a) Rock salt structure of bulk SnTe in a tetragonal conventional cell. $a$ is the lattice constant of the conventional cubic cell, and $b=c=a/\sqrt{2}$. Green planes are equivalent mirror planes. (b) Brillouin zone associated with the tetragonal cell, with its high-symmetry points indicated in red and the unique $M_z$-invariant plane in green. The projected 2D Brillouin zone with its high-symmetry points is shown on top.

Figure 5

(a) Energy bands of bulk SnTe along high-symmetry lines of the folded tetragonal BZ. The Fermi level is indicated by the dashed line. (b) Wannier band structure obtained from the full set of valence states. (c) Detail of the Wannier bands around the $z=0$ mirror plane. (d) Heat map plot of the gap function of Eq. (52) for the central pair of Wannier bands around $z=0$, with the nodal points color coded as in Fig. 3.

Figure 6

Topological phase diagram of the model of Eq. (54) for $c=1.0$. Orange and blue regions denote axion-even ($\theta =0$) and axion-odd ($\theta =\pi$) phases, respectively.

Figure 7

(a) Energy bands of the model described by Eq. (54) with $c=m=1.0$ and $M=0.5$. The bands are doubly degenerate, and the Fermi level (dashed line) has been placed at midgap. (b) Wannier band structure obtained from the valence states. (c) and (d) Heat map plots of the gap function of Eq. (52) about the $z=0$ and $z=c/2$ planes, respectively, with the nodal points color coded as in Fig. 3.

Figure 8

(a) Energy bands of the same model as in Fig. 7, after adding an extra pair of occupied orbitals with $E=-4.0$ at $z=\pm 0.2c$ and coupling them to the other orbitals. The bands are doubly degenerate, and the Fermi level (dashed line) has been placed at midgap. (b) Wannier band structure obtained from the valence states, with small gaps around $z=\pm 0.2c$ due to the added orbitals.

Figure 9

(a) Energy bands of the same model as in Fig. 7, after adding an extra occupied orbital at $z=0$ and coupling it to the other orbitals. The Fermi level (dashed line) has been placed in the gap. (b) Wannier band structure obtained from the valence states. The added orbital generates a flat band at $z=0$, which repels the nodal points on that plane (lower panel).

Figure 10

Wannier bands of the modified Dirac model on a cubic lattice [Eq. (54)], for $m=1.0$ and varying $M$.