Surfaces of axion insulators

Axion insulators are magnetic topological insulators in which the nontrivial ${\mathbb{Z}}_2$ index is protected by inversion symmetry instead of time-reversal symmetry. The naturally gapped surfaces of axion insulators give rise to a half-quantized surface anomalous Hall conductivity (AHC), but the sign of the surface AHC cannot be determined from topological arguments. In this paper, we consider topological phenomena at the surface of an axion insulator. To be explicit, we construct a minimal tight-binding model on the pyrochlore lattice and investigate the all-in-all-out (AIAO) and ferromagnetic (FM) spin configurations. We also implement a recently proposed approach for calculating the surface AHC directly, which allows us to explore how the interplay between surface termination and magnetic ordering determines the sign of the half-quantized surface AHC. In the case of AIAO ordering, we show that it is possible to construct a topological state with no protected metallic states on boundaries of any dimension (surfaces, hinges, or corners), although chiral hinge modes do occur for many surface configurations. In the FM case, rotation of the magnetization by an external field offers promising means of control of chiral hinge modes, which can also appear on surface steps or where bulk domain walls emerge at the surface.

Figure 1

(a) The pyrochlore lattice, comprised of an fcc lattice with a four-point basis (denoted by the numbers 1–4), forming a corner sharing tetrahedral network. (b) All-in-all-out configuration predicted for pyrochlore iridates. (c) The Ir $5d$ level splitting in pyrochlore iridates due to the crystal field and the spin-orbit coupling. (d) Graphical definition of ${\widehat{\mathit{b}}}_{ij}$ and ${\widehat{\mathit{d}}}_{ij}$

Figure 2

(a) The first Brillouin zone with the path along which the bulk energy bands are calculated. (b)–(d) Energy bands for different values of the parameters $(t,\lambda ,\mathrm{\Delta })$. (b) $(1,0,0)$. (c) $(1,0.1,0)$. (d) $(1,0.1,0.2)$.

Figure 3

Phase diagram for $t=1.0$ with the distribution of the total number of odd parity states at the TRIM. The Hamiltonian remains invariant under $\mathrm{\Delta }\to -\mathrm{\Delta }$ so it is sufficient to consider positive $\mathrm{\Delta }$. On the other hand, $H(-t,-\lambda ,\mathrm{\Delta })=-H(t,\lambda ,\mathrm{\Delta })$ so for $t=-1.0$ the occupied and unoccupied bands are interchanged. The phase diagram is nevertheless the same, since we assume half filling and the 8 bands constitute a trivial insulator.

Figure 4

Surface band structures for [(a) and (b)] $\left(111\right)$ and [(c) and (d)] $\left(001\right)$ surfaces. Blue (red) bands are localized on the bottom (top) surface. [(a) and (c)] Strong TIs, $H(t,\lambda ,\mathrm{\Delta })=(1,0.1,0)$, with protected metallic surface states. [(b) and (d)] Axion insulators, $H(t,\lambda ,\mathrm{\Delta })=(1,0.1,0.4)$; note gapped surface states on the (111) surface.

Figure 5

(a) Layer-resolved partial Chern number $C\left(l\right)$ of Eq. (15) as a function of layer depth $l$ for slabs along $\left(111\right)$ and $\left(001\right)$ directions. For both slabs the bulk parameters are $(t,\lambda ,\mathrm{\Delta })=(1.0,0.1,0.4)$ corresponding to the axion phase. In the case of the $\left(001\right)$ direction, the surface Zeeman term is modified to ${\mathrm{\Delta }}_{\mathrm{surf}}=0.8$ in order to gap the surface states. (b) Integral of $C\left(l\right)$ over a surface region extending to depth $n$ as computed from Eq. (24). [(c) and (d)] Sketches of (001) and (111) slab orientations respectively for the AIAO spin configuration.

Figure 6

WCCs for an axion insulator along the 4 projected TRIM for Wannierization directions (111) [(a) and (b)] and (001) [(c),(d)]. (a) and (c) correspond to a region in the phase diagram with 10 odd parity states at the TRIMs, where (b) and (d) correspond to 14 odd parity states.

Figure 7

Gapped surface states in the FM axion-insulator phase. Slabs along (a) (111) and (b) (001) directions. In both cases, the magnetization is perpendicular to the surface.

Figure 8

Surface AHC for a $\left(111\right)$ slab as the magnetization is rotated: (a) in a plane perpendicular to the surface; (b) in the plane of the surface. The polar angle $\vartheta =0$ corresponds to the $\left[\overline{1}\overline{1}\overline{1}\right]$ direction while the azimuthal angle $\phi =0$ corresponds to the $\left[01\overline{1}\right]$ direction ($\left[01\overline{1}\right]$ is perpendicular to one of the mirror planes).

Figure 9

Illustration of possible surface AHC configurations for an AIAO insulator in the axion phase with (001)-type surface terminations. Blue and red colors represent positive and negative surface AHC respectively as defined in an outward-directed convention. (a) Removal of one atomic layer creates a chiral step channel. (b) Intersection of a domain wall and a chiral step channel results in a junction with two incoming and two outgoing chiral channels. (c) Intersection of facets with different signs of surface AHC generates chiral channels along the hinges. (d) Manipulation of surface terminations by addition or removal of layers can eliminate edge channels and lead to a configuration displaying the topological magnetoelectric effect (see text).

Figure 10

Higher-order topological states at the surface of an FM axion insulator. (a) Octahedral crystallite with facets perpendicular to the (111) and equivalent directions. (b) Higher-order phase diagram. A given magnetization vector intersects the cuboctahedron at a particular face, with each face corresponding to a different arrangement of signs of the surface AHC on the crystallite. [(c)–(e)] Examples of surface configurations resulting from different magnetization directions. Color scheme follows that of Fig. 9.