Magnetic doping-induced second-order and first-order topological phase transition inthe photonic alloy

Abstract

The bulk-edge correspondence principle, a cornerstone of topological physics, ensures that first-order topological systems host robust chiral edge states in two dimension. This was later extended to higher-order phases, where second-order topological insulators exhibit localized, topologically protected corner states. While the transition between these distinct phases has been demonstrated in periodic systems, its existence in disordered platforms remains an open question. Here, we demonstrate a controllable topological phase transition between a second-order topological phase and a first-order topological phase in a two-dimensional photonic alloy. By tuning the magnetic doping concentration - implemented by attaching permanent magnets randomly to nonmagnetized yttrium iron garnet rods in an alternately magnetized honeycomb lattice with C3 rotational symmetry - we flexibly control the system's topology. At zero doping, we observe higher-order corner states, confirmed by a trivial Chern number and non-zero bulk polarizations of 1/3. As doping concentration increases, these corner states progressively merge with the bulk states, culminating in the closure of the bulk transmission gap. After the bulk transmission gap reopens with further increased doping, the system transitions to a first-order topological phase, characterized by a nontrivial Chern number of -1 and the emergence of a chiral edge state. This transition is reversible, providing a highly tunable and experimentally simple platform for flexibly switching between localized corner states and delocalized chiral edge states within a single photonic system.

Figures (6)

• Figure 1: Band structures of honeycomb lattice photonic crystals with two different symmetries: $C_{3}$ rotational symmetry ($C_{3}$ lattice, with YIG rods alternately magnetized) and $C_{6}$ rotational symmetry ($C_{6}$ lattice, with all YIG rods magnetized). (a), (e) Schematics of the $C_{3}$ and $C_{6}$ lattices, respectively. Black (red) circles denote magnetized (nonmagnetized) YIG rods. The dashed rhombi represent the primitive cells of the two lattices, and sites $A$ and $B$ correspond to two distinct atoms in the primitive cells. $a$ represents the lattice constant and $r$ is the radius of the YIG rods. (b), (f) Band structures of the $C_{3}$ and $C_{6}$ lattices, respectively. The yellow and pink shaded regions denote the first photonic band gaps of the $C_{3}$ and $C_{6}$ lattices, with their associated Chern numbers labeled. (c)-(d) Normalized $E_{{z}}$ field distributions of the eigenmodes at the $K$ point for the first and second bands of the $C_{3}$ lattice, respectively. These eigenmodes are indicated by the purple upward and red downward triangles in panel (b). (g)-(h) Normalized $E_{{z}}$ field distributions of the eigenmodes at the $K$ point for the first and second bands of the $C_{6}$ lattice, marked by the red upward and purple downward triangles in panel (f). In both symmetry configurations, the lattice constant is set to $a = 20\ mm$ and the radius of each YIG rods is $r = 2\ mm$.
• Figure 2: Topologically protected localized corner states and the delocalized chiral edge state realized within a photonic alloy. (a) Schematic of the photonic alloy structure, constructed by randomly replacing some nonmagnetized YIG rods (red dots) with magnetized YIG rods (black dots) in a $2$D alternately magnetized honeycomb lattice photonic crystal with $C_{3}$ rotational symmetry [see Fig. \ref{['fig:1']}(a)]. The brown boundary represents the perfect electric conductor (PEC). Colored stars labeled with numbers $1$ and $2$ denote the positions of the source and probe antennas used to obtain corner, edge, and bulk transmissions, respectively. (b)-(c) Normalized electric field distributions $\left | E_{{z}} \right |$ under line sources excitation for doping concentrations $x_{c} = 0.1$ and $x_{c} = 0.7$, respectively. Hollow white circles represent magnetized YIG rods, green dots represent nonmagnetized YIG rods, the blue stars indicate the locations of the line sources, and the blue arrow shows the propagation direction of the chiral edge state. The upper boundary in (c) uses an absorbing boundary condition, while the remaining boundaries are PEC, the same as in (a).
• Figure 3: Bulk transmission spectrum as a function of doping concentration $x_{c}$. Each data point represents the average of twenty disorder samples. The red dots in the high-doping region mark the boundary of the topological gap characterized by a Chern number of $-1$, as determined using the scattering method. Inset: Schematic of numerically calculated transmission in the photonic alloy system. The top and bottom boundaries are continuously connected, while the left and right boundaries are enclosed using absorbing boundary conditions. The location of the line source is marked by a blue star.
• Figure 4: Frequency-resolved transmissions and field distributions for the photonic alloy. (a), (e), (i) Frequency-resolved transmissions of corner, edge, and bulk states at doping concentrations $x_{c} =$$0$, $0.1$, and $0.2$, respectively. The colors of the transmission peaks correspond to the colors of the source-probe pairs used in Fig. \ref{['fig:2']}(a) to excite and collect information about these states. (b)-(d) Distributions of the absolute electric field value ($\left | E_{{z}} \right |$) for the three topological corner states excited in order of increasing frequency when the photonic alloy's doping concentration $x_{c} = 0$ ($C_{3}$ lattice). (f)-(h), (j)-(l) Field distributions showing the localization of corner states that evolved from the three topological corner states in (b)-(d) at doping concentrations $x_{c} = 0.1$ and $x_{c} = 0.2$. Blue stars mark the line source locations in (b)-(d), (f)-(h), and (j)-(l). Other simulation parameters are identical to those used in Figs. \ref{['fig:2']}(b)-\ref{['fig:2']}(c).
• Figure 5: Characterization of topological behavior in photonic alloys using the scattering method. (a) Schematic of the setup for extracting topological signatures from a hexagonal photonic alloy. The photonic alloy has a twisted boundary condition $\psi$ (y = L) =$\psi$ (y = 0)$e^{i\theta }$ applied to its top ($\textit{y} = L$) and bottom ($\textit{y} = 0$) boundaries. The left side connects to an air lead bounded by a perfect magnetic conductor, while the right boundary has an absorbing condition. A TM-polarized incident wave $E_{{z}}^{+}$ enters the system, and the reflected waveguide modes $E_{{z}}^{-}$ are analyzed. (b) Reflection phase $\phi$ plotted against twisting angle $\theta$ for different doping concentrations. Black, blue, and red hollow circles represent doping concentrations of $x_{c} =$$0.3$, $0.7$, and $0.9$, respectively.