Multimode One-Way Waveguides of Large Chern Numbers

Current experimental realizations of the quantum anomalous Hall phase in both electronic and photonic systems have been limited to a Chern number of one. In photonics, this corresponds to a single-mode one-way edge waveguide. Here, we predict quantum anomalous Hall phases in photonic crystals with large Chern numbers of 2, 3, and 4. These new topological phases were found by simultaneously gapping multiple Dirac and quadratic points. We demonstrate a continuously tunable power splitter as a possible application of multimode one-way waveguides. All our findings are readily realizable at microwave frequencies.

Figure 1

Bulk and edge TM band structures showing two one-way edge states ($C_{\text{gap}}=2$) obtained from four Dirac points. The photonic crystal is a square lattice of rods with a radius of $0.13a$, $\epsilon =13$, and $\mu =1$. The $T$-breaking perturbation corresponds to adding ${\mu }_{12}=-{\mu }_{21}=0.40i$ to the rods. (a) Bulk band structure showing the Dirac point along $M-K$. The lower inset illustrates the lattice geometry. (b) Four Dirac cones between the fourth and fifth bands plotted in the whole Brillouin zone. (c) Bulk band structure under $T$-breaking perturbation opens a 5.5% complete gap highlighted in yellow. Each band is labeled with its Chern number. (d) Two gapless one-way edge states (red lines) appear in the projected edge band diagram when the bulk is terminated by a metallic boundary.

Figure 2

Bulk and edge TM band structures showing four one-way edge states ($C_{\text{gap}}=4$) obtained from four Dirac points and two quadratic points. The photonic crystal is a square lattice of rods with a radius of $0.07a$, $\epsilon =24$, and $\mu =1$. The $T$-breaking perturbation corresponds to adding ${\mu }_{12}=-{\mu }_{21}=0.40i$ to the rods. (a) Bulk band structure showing the Dirac point along $\mathrm{\Gamma }-M$ and two quadratic points at $\mathrm{\Gamma }$ and $K$, respectively. The lower inset illustrates the lattice geometry. (b) Four Dirac cones and two quadratic points between the sixth and seventh bands plotted in the whole Brillouin zone. (c) Bulk band structure under $T$-breaking perturbation opens a 3.0% complete gap highlighted in yellow. Each band is labeled with its Chern number. The Chern number of degenerate bands is circled by a dotted line. (d) Four gapless one-way edge states (red lines) appear in the projected edge band diagram when the bulk is terminated by a metallic boundary. (e) Mode profiles of the four edge modes close to the metallic interfaces are plotted at the same frequency dotted in (d).

Figure 3

The TM topological gap map of a photonic crystal consisting of a square lattice of YIG rods under an applied static magnetic field. The gap Chern number labels each complete band gap found. The previously reported single-mode one-way waveguides correspond to the blue band gap of $C_{\text{gap}}=-1$ [3, 15] and the green band gap of $C_{\text{gap}}=1$ [21]. The left lower inset illustrates the lattice geometry with a period of $a$. The upper right inset illustrates a 2D one-way circuit by interfacing various photonic crystals of different gap Chern numbers.

Figure 4

A power splitter implemented with $C_{\text{gap}}=2$ and $C_{\text{gap}}=1$ gyromagnetic photonic crystals bordered on the top by a metallic wall. A point source with frequency $a/\lambda =0.82$ couples light into the multimode waveguide to the left of the metal scatterer. The unit cell size of the $C_{\text{gap}}=2$ lattice is $a$, the unit cell size of the $C_{\text{gap}}=1$ lattice is $0.805a$, and the rod radius for both lattices is $0.15a$. The operating parameters are the same as in Fig. 3, so $a$ in real units is 1.7 cm. (a) For a metal scatterer with height $1.65a$, the majority of the light proceeds through channel 2 (bottom waveguide). (b) For a metal scatterer with height $1.30a$, the majority of the light proceeds through channel 1 (right waveguide). (c) The transmission to each waveguide as a function of metal scatterer height.