Surface theorem for the Chern-Simons axion coupling

The Chern-Simons axion coupling of a bulk insulator is only defined modulo a quantum of $e^2/h$. The quantized part of the coupling is uniquely defined for a bounded insulating sample, but it depends on the specific surface termination. Working in a slab geometry and representing the valence bands in terms of hybrid Wannier functions, we show how to determine that quantized part from the excess Chern number of the hybrid Wannier sheets located near the surface of the slab. The procedure is illustrated for a tight-binding model consisting of coupled quantum anomalous Hall layers. By slowly modulating the model parameters it is possible to transfer one unit of Chern number from the bottom to the top surface over the course of a cyclic evolution of the bulk Hamiltonian, changing the surface anomalous Hall conductivity by a quantum of conductance $e^2/h$. When the evolution of the surface Hamiltonian is also cyclic, the Chern pumping is obstructed by chiral touchings between valence and conduction surface bands.

Figure 1

Sketch of the slab configuration discussed in the text (adapted from Ref. [19]). The HWFs are localized in the surface-normal direction $\widehat{\mathit{n}}=\widehat{\mathit{z}}$.

Figure 2

(a) Four unit cells of an isolated haldanium layer. Sites on the $A$ $\left(B\right)$ sublattices are marked with open (filled) circles, and the arrows indicate the directions of the second-nearest-neighbor hoppings in Eq. (55) with amplitudes $i{(-1)}^p{t}_2$. (b) First Brillouin zone, with high-symmetry points marked. (Reproduced from Ref. [31].) When viewing (b) as the projected BZ of a layered 3D model in Sec. 3b, the labels of the high-symmetry points become $\overline{\mathrm{\Gamma }},\overline{\mathrm{M}},\overline{\mathrm{K}}$, and ${\overline{\mathrm{K}}}^{\prime }$.

Figure 3

Numerical results for the bilayer model as a function of the adiabatic loop parameter $\varphi$. Top: Minimum energy gap between the second (highest occupied) and third bands; the minimum gap is always at point K in the 2D BZ. Middle: Wannier charge centers at K, in units of the layer separation $c/2$. A dashed line indicates a Wannier sheet with Chern number $C=0$, a heavy (red) solid line denotes $C=+1$, and a light (green) solid line denotes $C=-1$. The lower (upper) sheet has index $n=1$ $\left(n=2\right)$. Bottom: Dimensionless CSA coupling ${\theta }_2/c$. Open circles, extrapolation of ${\theta }_2\left(N\right)/c$ from finite-size samples. Solid line, direct calculation using the $k$-space formula (37).

Figure 4

Numerical results for the bulk model as a function of the adiabatic loop parameter $\varphi$. Top: Minimum energy gap between the second and third bands; the minimum gap is always at point H, which projects onto $\overline{\mathrm{K}}$ in the surface BZ. Middle: Periodically repeated Wannier charge centers at $\overline{\mathrm{K}}$, in units of the lattice constant $c$; the dashed line and colored heavy/light lines have the same meaning as in Fig. 3. The lower Wannier sheet has indices $l=-1$ and $n=2$, the middle one $l=0$ and $n=1$, and the upper one $l=0$ and $n=2$. Bottom: Total CSA coupling ${\theta }_3={\theta }_{z\mathrm{\Omega }}+{\theta }_{\mathrm{\Delta }xy}$, and the term ${\theta }_{z\mathrm{\Omega }}$ responsible for pumping.

Figure 5

Numerical results for a ten-layer slab $\left(N_z=5\right)$ as a function of the adiabatic loop parameter $\varphi$. The surface Hamiltonian has been modified according to Eq. (62), to keep the surface insulating throughout the cycle. Top: Minimum energy gap (always at point K in the 2D BZ) at half filling. Middle: Wannier charge centers at K in units of the lattice constant $c$, with indices ranging from $n=1$ (bottom) to $n=10$ (top); the dashed line and colored heavy/light lines have the same meaning as in Fig. 3. Two possible choices of bulk cells are indicated by the gray and black lines. Bottom: Dimensionless CSA coupling ${\theta }_{\mathrm{slab}}={\theta }_2/N_{z}c$. Heavy line, calculation done at $N_z=5$. Open circles, extrapolation to $N_z\to \infty$ of calculations done at $N_z=4,7,10$. Light gray line, bulk CSA coupling ${\theta }_3$ taken from Fig. 4.

Figure 6

Same as the top and bottom panels of Fig. 5, except that now the entire slab, including the surfaces, undergoes a cyclic evolution.

Figure 7

Surface spectral function in a semi-infinite geometry, calculated at the critical parameter value ${\varphi }_c=3\pi /2$ where the CSA coupling in Fig. 6 changes discontinuously (top panel), and at $\varphi ={\varphi }_c\pm \pi /20$ (bottom panel). The spectral function is plotted along two lines that extend $(1/20)\mathrm{th}$ of the distance from $\overline{\mathrm{K}}$ to $\overline{\mathrm{M}}$ and $\overline{\mathrm{\Gamma }}$.

Figure 8

Surface spectral function of the layered model at $\varphi =0$ and $\varphi =\pi$, for a semi-infinite geometry with the surface orthogonal to the $y$ direction. In both cases the entire system (bulk plus surface) has mirror symmetry, but while in the top panel the bulk CSA coupling vanishes, in the bottom panel it equals $\pi$ and protects a surface Dirac cone.

Figure 9

Top: Band structures in the 3D BZ, calculated at $\varphi =3\pi /2$ with three different values of $\beta$. The bands are everywhere twofold degenerate, and at $\beta =\pi /6$ they become fourfold degenerate at H, forming a Dirac point. Bottom: Interlayer couplings, hybrid Wannier charge centers at point $\overline{\mathrm{K}}$ in the projected BZ, and bulk CSA coupling, plotted as a function of $\varphi$ for the same three values of $\beta$. In the panels depicting the Wannier centers, the dashed lines and colored heavy/light lines have the same meaning as in Fig. 3.