Hinge states of second-order topological insulators as a Mach-Zehnder interferometer

Three-dimensional higher-order topological insulators can have topologically protected chiral modes propagating on their hinges. Hinges with two copropagating chiral modes can serve as a “beam splitter” between hinges with only a single chiral mode. Here we show how such a crystal, with Ohmic contacts attached to its hinges, can be used to realize a Mach-Zehnder interferometer. We present concrete calculations for a lattice model of a first-order topological insulator in a magnetic field, which, for a suitable choice of parameters, is an extrinsic second-order topological insulator with the required configuration of chiral hinge modes.

Figure 1

(a) A schematic of the system considered in this article. It consists of a second-order topological insulator with hinges that have one or two chiral states. Ohmic contacts, which pairwise serve as source (S1, S2) and drain (D1, D2) contacts, are attached to four of the hinges with one chiral mode. A pair of interfering paths connecting contacts S1 and D1 is indicated in blue. (b) Effective network diagram indicating the same interfering paths between contacts S1 and D1.

Figure 2

(a) An extrinsic second-order topological insulator with chiral hinge modes can be realized by placing a first order topological insulator in a uniform magnetic field ${\mathbf{B}}_0$. The direction of ${\mathbf{B}}_0$ is chosen such that there is a nonzero flux through all crystal surfaces. As a result, the Dirac-cone surface states form an integer quantized Hall effect with a filling fraction that depends on the magnetic flux density and the electron density at the surface. The latter can be controlled capacitively by a metal gate, one is shown explicitly on surface $x+$. (b) Labeling of surfaces and hinges. The surfaces on the sides are labeled by $x\pm$ and $y\pm$ analogously to $z\pm$, labeled here in red.

Figure 3

Band structure of a lattice infinite along the $x$ axis (top), $y$ axis (middle), and $z$ axis (bottom) and with a finite size in the other two coordinate directions. The flat bands correspond to the surface Landau levels. The dispersing bands are hinge states. The hinge modes are labeled by which hinge they appear on, and bulk surface modes by their surface, following the convention of Fig. 2 (right panel). Parameter values are described in the main text. The lattice sizes in the directions in which they are finite are $L_x=50a, L_y=80a$, and $L_z=80a$. The horizontal dashed line indicates the Fermi energy ${\varepsilon }_{\mathrm{F}}=0.05t$. The zeroth Landau level for the $z-$ surface is at $\varepsilon =0.3t$, which is outside the range shown in the figure.

Figure 4

Effective network diagram for the Mach-Zehnder interferometer of Fig. 1 with labeling of the crystal edges following the convention of Fig. 2. The interfering paths from source contact S1 to drain contact D1 are indicated in blue. The edges with two copropagating hinge modes serve as the beam splitters, as indicated by the blue ovals.

Figure 5

Differential conductance $G$ versus magnetic flux $\mathrm{\Phi }=B_z{L}_x{L}_y$ for the Mach-Zehnder geometry of Fig. 1 (left) with $B_y=B_x=0$. Curves are shown for two different values of the Fermi energy ${\varepsilon }_{\mathrm{F}}$ and two different values of the system size $L_z$, as indicated in the figure.

Figure 6

Two-terminal Aharonov-Bohm interferometer con- structed from a second-order topological insulator with hinges that have one or two hinge states (left) and equivalent network diagram (right).

Figure 7

Differential conductance $G$ versus magnetic flux $\mathrm{\Phi }$ for the two-terminal geometry of Fig. 6 (red curves) and for the four-terminal geometry of Fig. 1 (orange curves). The three panels correspond to three directions of the magnetic field variation: $\delta \mathbf{B}\propto (cos\theta ,0,sin\theta )$, with $\theta =0$ (left), $\theta =0.2\pi$, (center), and $\theta =\pi /2$ (right). System parameters are $L_x=L_y=L_z=50a$ and ${\varepsilon }_{\mathrm{F}}=0.06t$. The magnetic flux $\mathrm{\Phi }$ is the sum of the fluxes through the three crystal faces, see Eq. (10).

Figure 8

Interference pattern of the differential conductance for an additional field $\delta \mathbf{B}\propto (cos\theta ,0,sin\theta )$. System parameters are $L_x=L_y=L_z=50a$ and ${\varepsilon }_{\mathrm{F}}=0.06t$.

Figure 9

Fourier transform of the differential conductance for an additional field $\delta \mathbf{B}\propto (cos\theta ,0,sin\theta )$. Results for the four-terminal (MZI, yellow) and two-terminal (AB, red) cases are shown. Dashed vertical black lines show the locations of the exact peaks in the Fourier transform. Parameter values are as in Fig. 7.

Figure 10

Full band structure in the bulk gap of a lattice infinite along the $x$ axis (top), $y$ axis (middle), and $z$ axis (bottom), and finite along the other two coordinate directions. The dispersing bands are hinge states. The flat bands correspond to the surface Landau levels. The bulk states are visible at the top and bottom of the plots. The parameter values are the same as in Fig. 3.

Figure 11

Geometry used to numerically obtain the single-corner scattering matrix $t_{z1}^{-}$ (left panel) and the beam-splitter scattering matrix $t_{z1}$ (right panel) using the kwant software. The red lattice sites represent the “ideal lead”, which is semi-infinite in one direction (only five layers shown). The color shows the support of a scattering state in the scattering region, at $\varepsilon =0.077t$. The blue arrows indicate the direction of propagation of the modes. The color scale has been saturated to make both of the outgoing hinge modes more visible. The interference pattern between the two co-propagating modes is clearly visible along the middle hinge in the bottom panel.

Figure 12

Off-diagonal beam-splitter transmission probability $|{\left(t_{z1}\right)}_{12}{|}^2$ as a function of the separation $L_z$ between corners at $\varepsilon =0.077t$. Blue triangle data points are obtained using the geometry of Fig. 11 bottom panel. Dashed lines and black circles are obtained from Eq. (6) with single-corner scattering matrices $t_{z1}^{\pm }$ obtained from a geometry as in Fig. 11 top panel.