Antihelical Edge States in Two-dimensional Photonic Topological Metals

Introduction.—The fundamental concepts in condensed matter physics introduced to topological photonics inspire the rapid development of photonic topological states LLu14 ; ABKNPo17 ; TORMP19 . The chiral edge states of topological insulators unidirectionally propagate along boundaries and require the breaking of time-reversal symmetry FHPRL08 ; ZWNa09 ; MHNP11 ; KFNP12 ; GQLPRL13 ; SZhangPRL20 . The degenerate clockwise and counterclockwise modes of ring resonators experience opposite artificial magnetic fields and provide a pseudospin degree of freedom MHNP11 . The helical edge states of time-reversal symmetric topological insulators with different pseudospins unidirectionally propagate in opposite directions. The edge states of topological metals are topologically-protected in the gapless phase. Antichiral edge states have been proposed and implemented on zigzag edges by modifying the next-nearest-neighbor hopping phase of the Haldane model FranzPRL18 ; ZhangBLPRL20 ; ChongYD21 ; LiZYPRL22 . Antichiral edge states propagate along the parallel lattice boundaries in the same direction. Recent progress in antichiral edge states has led to the development of a solution for creating topological metals. Topological metals are difficult to create because of the challenge of separating gapless bands and breaking time-reversal symmetry in photonics. Thus, is it possible to have time-reversal symmetric topological metals with topologically-protected edge states and robust propagation?

Here, we introduce the antihelical edge states [Fig. 1(a)] of time-reversal symmetric topological metals and a photonic realization is proposed in the two-dimensional anisotropic square lattice of coupled ring resonators. The pseudospins are time-reversal symmetric counterparts, and the introduction of pseudospins addresses the difficulty of breaking time-reversal symmetry in photonics. The symmetric component of next-nearest-neighbor couplings creates a nontrivial topology, and the anti-symmetric component of next-nearest-neighbor couplings separates energy bands and supports the antihelical edge states on both horizontal and vertical boundaries. Antihelical edge states robustly copropagate along corners with opposite pseudospins that separate and converge at the opposite corners on the diagonal of a lattice. The antihelical edge states become helical edge states during a metal-insulator phase transition [Fig. 1(b)]. Photonic topological metals provide a new direction for research on topological photonics.

Anisotropic square lattice of ring resonators.—Figure 1(c) presents a two-dimensional anisotropic square lattice of coupled ring resonators (see Supplementary materials A). The clockwise mode (pseudospin-up) and counterclockwise mode (pseudospin-down) are time-reversal counterparts, and they experience the opposite Peierls phases in the horizontal couplings between the nearest-neighbor resonators MHNP11 . The lattice for the pseudospin-up (pseudospin-down) is presented in Fig. 1(d) [Fig. 1(e)]. The horizontal couplings $\pm iJ_{1}$ indicated in green and $\pm iJ_{2}$ indicated in orange are tunneling-direction-dependent, break the time-reversal symmetry of the Hamiltonian for each individual pseudospin, and separately affect the edge states localized on the upper and lower boundaries. The vertical coupling between the nearest-neighbor resonators is $\kappa$ indicated in black. The cross couplings between the next-nearest-neighbor resonators are $\chi_{1}=\chi+\delta$, $\chi_{2}=\chi-\delta$ LeykamPRL18 . The symmetric component $\chi$ opens a band gap [Fig. 1(f)] and creates a nontrivial topology, whereas the anti-symmetric component $\delta$ affects the band structure, separates the bands in both the $x$ and $y$ directions, and creates antihelical edge states that are localized on the left and right boundaries in the topological metallic phase.

Topological phases.—The Bloch Hamiltonian for the pseudospin-up is $h_{\uparrow}\left(\mathbf{k}\right)=d_{0}\left(\mathbf{k}\right)\sigma_{0}+\mathbf{d\left(\mathbf{k}\right)\cdot\sigma}$, where $\sigma_{0}$ is the identical matrix and $\mathbf{\sigma=}\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)$ is the Pauli matrix. The first term adjusts the band energy and the second term determines the band topology. We have $d_{0}\mathbf{\left(\mathbf{k}\right)}=(J_{1}+J_{2})\sin k_{x}$ and $\mathbf{d}\left(\mathbf{k}\right)=\mathbf{r}_{1}(k_{x})-\mathbf{r}_{2}(k_{y})$ with $\mathbf{r}_{1}(k_{x})=(\kappa+2\chi\cos k_{x},-2\delta\sin k_{x},(J_{1}-J_{2})\sin k_{x})$, $\mathbf{r}_{2}(k_{y})=(-\kappa\cos k_{y},\kappa\sin k_{y},0)$. $h_{\uparrow}\left(\mathbf{k}\right)$ respects the particle-hole symmetry $\sigma_{z}h_{\uparrow}^{T}\left(\mathbf{k}\right)\sigma_{z}^{-1}=-h_{\uparrow}\left(-\mathbf{k}\right)$. The topological phase belongs to the class $D$ and it is characterized by a $\mathbb{Z}$ topological invariant, that is, the spin-Chern number $C_{\uparrow}=-\left(4\pi\right)^{-1}\int\left|\mathbf{d}\right|^{-3}\mathbf{d}\cdot\left(\partial_{k_{x}}\mathbf{d}\times\partial_{k_{y}}\mathbf{d}\right)dk_{x}dk_{y}$ [Fig. 1(g)]. The band topology is captured by the effective magnetic field $\mathbf{d}\left(\mathbf{k}\right)$. After $\mathbf{d}\left(\mathbf{k}\right)$ is substituted with $\mathbf{r}_{1}(k_{x})-\mathbf{r}_{2}(k_{y})$ in $C_{\uparrow}$, the spin-Chern number becomes the definition of a linking number of the two independent periodic vectors $\mathbf{r}_{1}(k_{x})$ and $\mathbf{r}_{2}(k_{y})$ (see Supplementary materials B). $\mathbf{r}_{1}(k_{x})$ is an ellipse that passes through $\left(\kappa-2\chi,0,0\right)$ and $\left(\kappa+2\chi,0,0\right)$. $\mathbf{r}_{2}(k_{y})$ is a circle centered at the origin with a fixed radius $\left|\kappa\right|$ in the $z=0$ plane. Thus, the two closed curves $\mathbf{r}_{1}(k_{x})$ and $\mathbf{r}_{2}(k_{y})$ are linked when $0<\left|\chi\right|<\left|\kappa\right|$.

In Fig. 2(a), the red circle $\mathbf{r}_{2}(k_{y})$ that is fixed on the $z=0$ plane is always clockwise; the rotation direction of the blue ellipse $\mathbf{r}_{1}(k_{x})$ reverses along the topological phase transition plane $J_{1}=J_{2}$, where the band gap vanishes and $\mathbf{r}_{1}(k_{x})$ becomes coplanar to $\mathbf{r}_{2}(k_{y})$. The blue ellipse $\mathbf{r}_{1}(k_{x})$ circles counterclockwise around the red circle $\mathbf{r}_{2}(k_{y})$ once in the region $J_{1}>J_{2}$ ($C_{\uparrow}=+1$); and the blue ellipse $\mathbf{r}_{1}(k_{x})$ circles clockwise around the red circle $\mathbf{r}_{2}(k_{y})$ once in the region $J_{1}<J_{2}$ ($C_{\uparrow}=-1$). The right-hand (left-hand) rule identifies $C_{\uparrow}=+1$ ($C_{\uparrow}=-1$): orient the thumb pointing along the arrow of the red circle, curl the rest of the fingers pointing along the arrow of the blue ellipse. Notably, in the phase diagram [Fig. 2(b)], two topological phases with opposite spin-Chern numbers have an identical number of edge states.

Antihelical edge states.—Topological edge states appear in the situation that the next-nearest-neighbor coupling $\chi$ is weak than the nearest-neighbor coupling $\kappa$. The nontrivial topology supports helical and antihelical edge states [Fig. 1(h)] that are different from their band structures and their ways of propagation. The antihelical (helical) edge states on the parallel boundaries propagate in the same (opposite) direction, being robust against disorder because of the topological protection and the spatial separation between the edge and bulk states FranzPRL18 .

In Fig. 1(b), the regions $J_{1}J_{2}<0$ are the topological insulating phase; the helical edge states with different pseudospins propagate clockwise or counterclockwise along the lattice boundaries. The topological phase undergoes a metal-insulator phase transition at $J_{1}J_{2}=0$. The regions $J_{1}J_{2}>0$ are the topological metallic phase KamenevPRL18 ; the edge states excited by the pseudospin-up propagate toward the left (or right) on the two parallel boundaries, but the edge states excited by the pseudospin-down propagate toward the right (or left) boundaries for $J_{1},J_{2}>0$ (or $J_{1},J_{2}<0$). These edge states associated with the two pseudospins constitute the antihelical edge states, and the required counterpropagating modes for the antihelical edge states are the bulk states. The bands touch at the topological phase transition, where topological edge states are degenerate and exhibit identical dispersion and propagation velocity.

The square lattice exhibits nontrivial topology throughout the parameter space $J_{1}$-$J_{2}$-$\delta$ [Fig. 2(a)]. The various types of edge states, as distinguished by their propagation, are presented in Fig. 2(b). The helical edge states are in $J_{1}J_{2}<0$ indicated in cyan and the antihelical edge states are in the other regions $J_{1}J_{2}>0$. The antihelical edge states differ from the helical edge states in term of their copropagation on the parallel boundaries MHNP11 ; FranzPRL18 . The antihelical edge state excitations with different pseudospins unidirectionally propagate in the opposite directions along the lattice boundaries and enable the separation of the robustly copropagated chiral mode. The helical edge states in the topological insulating phase always appear in both the horizontal and vertical directions; however, the antihelical edge states in the topological metallic phase usually appear only in one direction because of the inseparable band energy in the other direction. Here, the antihelical edge states are simultaneously present in both the horizontal and vertical directions for $\delta^{2}>J_{1}J_{2}>0$, where the energy bands are separable in both directions. The antihelical edge states are only present in the horizontal directions for $\delta^{2}<J_{1}J_{2}$. In Fig. 2(b), the surfaces $J_{1}J_{2}=0$ and $\delta^{2}=J_{1}J_{2}$ divide the parameter space $J_{1}$-$J_{2}$-$\delta$ into three phases with eight regions. The eight types of edge state propagation for the pseudospin-up excitations are presented in Fig. 2(c). The pseudospin-down edge state excitations propagate in the opposite directions.

The simultaneous sign changes of $J_{1}$ and $J_{2}$ reverse the propagation directions of all the antihelical edge states. The four cases of edge modes on the left-half and right-half of Fig. 2(c) propagate in opposite directions. The sign change of $\delta$ only alters the propagation direction of the antihelical edge states in the vertical direction. The antihelical edge states at the strong $\chi_{1}$ ($\delta>0$) pass along the bottom-left and top-right corners and scatter into the bulk at the top-left and bottom-right corners; by contrast, the antihelical edge states at the strong $\chi_{2}$ ($\delta<0$) pass along the bottom-right and top-left corners and scatter into the bulk at the top-right and bottom-left corners. In Fig. 2(c), the helical edge states propagate counterclockwise (clockwise) along the lattice boundaries for $C_{\uparrow}=+1$ ($C_{\uparrow}=-1$) in the cyan region $J_{1}>0,J_{2}<0$ ($J_{1}<0,J_{2}>0$). The antihelical edge states along the horizontal direction in the red region $J_{1},J_{2}>0$ ($J_{1},J_{2}<0$) of $\delta^{2}<J_{1}J_{2}$; the edge state excitation propagates leftward (rightward), scatters into the bulk at the corners, and goes backward. The antihelical edge states are present along both the horizontal and vertical directions when $0<J_{1}J_{2}<\delta^{2}$; the edge state excitation propagates half a closed-loop along the lattice boundaries, scatters into the bulk at the corners, and goes backward along the diagonal direction with the support of the scattering states. The four cases are respectively distributed in the orange regions $\left(J_{1},J_{2},\delta>0\right)$, $\left(J_{1},J_{2}<0,\delta>0\right)$, $\left(J_{1},J_{2}>0,\delta<0\right)$, and $\left(J_{1},J_{2},\delta<0\right)$ of $0<J_{1}J_{2}<\delta^{2}$.

Figures 2(d)-2(g) present the four cases of the band structures for the pseudospin-up presented on the left-half of Fig. 2(c). The propagation of antihelical edge states in Figs. 2(f) and 2(g) for the pseudospin-up are simulated in Figs. 2(h) and 2(i), and the corresponding propagation for the pseudospin-down are simulated in Figs. 2(j) and 2(k). The separation and convergence of different pseudospins toward the opposite corners of the square lattice are observed.

In conclusion, we have introduced the antihelical edge states in the time-reversal symmetric topological metallic phase, and proposed the photonic realization in coupled ring resonators. The antihelical edge states with different pseudospins propagate in the opposite directions along the corners toward the diagonal of the square lattice. Unconventional copropagation is applicable for the robust spin purification. The antihelical edge states become the helical edge states after undergoing a metal-insulator phase transition. The wide range of reconfigurable robust light propagation enables the flexible control of light flow at the edges. Our findings provide insight into the photonic topological metals and are applicable for the acoustic lattices and other two-dimensional metamaterials. Concepts pertaining to topological metals and antihelical edge states are inspiring in the condensed matters and topological materials.