Gapless hinge states from adiabatic pumping of axion coupling

We demonstrate that chiral hinge modes naturally emerge in insulating crystals undergoing a slow cyclic evolution that changes the Chern-Simons axion angle $\theta$ by $2\pi$. This happens when the surface (not just the bulk) returns to its initial state at the end of the cycle, in which case it must pass through a metallic state to dispose of the excess quantum of surface anomalous Hall conductivity pumped from the bulk. If two adjacent surfaces become metallic at different points along the cycle, there is an interval in which they are in topologically distinct insulating states, with chiral modes propagating along the connecting hinge. We illustrate these ideas for a tight-binding model consisting of coupled layers of the Haldane model with alternating parameters. The surface topology is determined in a slab geometry using two different markers, surface anomalous Hall conductivity and surface-localized charge pumping (flow of surface-localized Wannier bands), and we find that both correctly predict the appearance of gapless hinge modes in a rod geometry. When viewing the axion pump as a four-dimensional crystal with one synthetic dimension, the hinge modes trace Fermi arcs in the Brillouin zone of the two-dimensional hinge connecting a pair of three-dimensional surfaces of the four-dimensional crystal.

Figure 1

(a) Real-space view of a sample of an insulating material undergoing an axion pumping cycle parametrized by $\varphi$. The sample is obtained by terminating a bulk crystal at two semi-infinite surfaces oriented normal to $+\widehat{\mathbf{z}}$ and $-\widehat{\mathbf{y}}$ (blue and red shadings, respectively), meeting at an $x$-directed hinge that may harbor chiral modes (arrows). (b)–(d) Hinge-projected band structure focusing on surface states on the $+\widehat{\mathbf{z}}$-oriented surface (blue shading), for three increasing values of $\varphi$. Gray shading represents projected bulk states. (e)–(g) View in $(k_x,k_y,\varphi )$ space, for the choice of Fermi level indicated in (b)–(d); blue indicates the “Fermi surface” in $(k_x,k_y,\varphi )$ space enclosing the electron pocket; the solid dot is the Weyl point, corresponding to the nodal touching of surface valence and conduction bands at the critical parameter value ${\varphi }_2$. The dashed lines in (e)–(g) indicate the $\varphi$ values used in (b)–(d), respectively. (h, i) Same as (d) and (g), but focusing on states localized on the surface normal to $-\widehat{\mathbf{y}}$.

Figure 2

(a) Hinge-projected Fermi surface of the system depicted in Fig. 1, plotted in $(k_x,\varphi )$ space. Blue and red electron pockets correspond to surface states on the $+\widehat{\mathbf{z}}$- and $-\widehat{\mathbf{y}}$-oriented surfaces, as shown in Figs. 1 and 1, respectively. The Fermi-arc state is shown in green. (b) Hinge-projected band-structure plot vs $k_x$ [at $\varphi ={\varphi }_3$, indicated by the dashed line in (a)]. The gray, blue, and red regions are the projected bulk, $+\widehat{\mathbf{z}}$-surface, and $-\widehat{\mathbf{y}}$-surface states shown in Figs. 1 and 1, and the green line is the chiral hinge state.

Figure 3

Pumping of the axion angle $\theta$ by $2\pi$ in the alternating Haldane model.

Figure 4

Left and middle: Inequivalent $z$-terminated slabs of the alternating Haldane model. Arrows indicate edge-mode chiralities on the uncoupled layers for $\pi /2<\varphi <3\pi /2$. Right: One layer of a $y_0$-terminated slab, with zigzag edges (top view).

Figure 5

(a) Evolution of the minimum energy gap vs $\varphi$ in slabs of the alternating Haldane model, for the three types of surface terminations pictured in Fig. 4. The gap closures occur at the surfaces. (b) Evolution of the surface AHC of each slab. Dotted lines are different branches of the bulk axion coupling, plotted as $-\theta /2\pi$ according to Eq. (10). The finite slopes of the discrete jumps at $\varphi =\pi /2, \pi$, and $3\pi /2$ are artifacts of the finite step size used for $\varphi$ in the calculation.

Figure 6

Topological phase diagram for 1D channels in the alternating Haldane model at half filling. The outer and middle racetracks are for the two types of $y|z$ hinges, and the inner one is for single-layer-high steps on $z$-oriented surfaces. In the yellow regions there are no protected 1D modes because the surface-AHC difference in Fig. 5 is $\mathrm{\Delta }n=0$, while in the blue ($\mathrm{\Delta }n=+1$) and red ($\mathrm{\Delta }n=-1$) regions there is one protected mode per hinge or step. Red, blue, and green lines mark the gap-closing points ${\varphi }_c$ on the $y_0, z_0$, and $z_{1/2}$ surfaces, respectively, that separate the different phases.

Figure 7

(a) Energy bands of the alternating Haldane model, calculated at $\varphi =5\pi /4$ for a rod extended along $x$ and with $y_0$ and $z_{1/2}$ terminations along $y$ and $z$. All bands are doubly degenerate, and those in red and in blue are hinge-localized chiral modes crossing the bulk and surface gaps, depicted schematically in the inset. (b) Site-resolved weights of the four hinge-localized states at an energy slightly above the crossing point in the middle of the gap, as indicated by the blue and red dots in (a).

Figure 8

(a) Fermi arcs traced on the $(k_x,\varphi )$ plane by the gapless hinge modes of a rod extended along $x$ and with a $z_{1/2}$ vertical termination, for the Fermi level at $E_{\mathrm{F}}=0$. (b) Same, but for $E_{\mathrm{F}}=0.2.$ The two elliptical discs indicate approximately the regions where the surface conduction bands move below $E=0.2$.

Figure 9

Wannier bands $z_{ln}\left(k_x\right)$ of $y$-terminated slabs of the alternating Haldane model, at different $\varphi$ values. The bands are color coded according to the degree of localization on the $+\widehat{\mathbf{y}}$ surface [Eq. (12)]: gray dots are modes extending along $y$ across the entire slab, and blue (red) dots are modes localized on the $+\widehat{\mathbf{y}}$ ($-\widehat{\mathbf{y}}$) surface; the degree of surface localization is also indicated by the size of the dots. Two types of cells are displayed in each panel: the “$z_0$ cell” with boundaries at $z=0\text{mod}\text{1}$ (in black), and the “$z_{1/2}$ cell” with boundaries at $z=1/2\text{mod}\text{1}$ (in gray).

Figure 10

Same as Fig. 9, but for the Wannier bands $y_{ln}\left(k_x\right)$ of $z_{1/2}$-terminated slabs.

Figure 11

Pumped charge ${\mathcal{P}}_z\left(k_x\right)$ of Eq. (13), in units of $e$, for a $y$-terminated slab at $\varphi =5\pi /4$. Blue (red) curves denote results for the $+\widehat{\mathbf{y}}$ ($-\widehat{\mathbf{y}}$) surface. In (a) and (b), Eq. (13) is evaluated using the $z_0$ and $z_{1/2}$ cells shown in Fig. 9, respectively. Filled dots correspond to the home cell [[0,1] in (a) and $[-1/2,1/2]$ in (b)], and open dots correspond to the cells immediately above and below.