Coupled-wire construction of non-Abelian higher-order topological phases

TL;DR

This paper develops a coupled-wire framework to realize non-Abelian higher-order topological phases, combining 1D non-Abelian chains with Abelian boundary topology. The minimal 2D model yields corner states only when the quaternion charge $Q_x\in\mathbb{Q}_8$ and winding $w\in\mathbb{Z}$ are simultaneously nontrivial, summarized by the hybrid invariant $\nu=(Q_x,w)$. A bulk-edge-corner correspondence is established, predicting corner-state degeneracies and the emergence of non-Abelian weak edge bands that couple to the 2D bulk. The approach enables experimental prospects in photonic/acoustic metamaterials and transmission-line networks and points to further extensions to non-Hermitian and Floquet HOTPs.

Abstract

Non-Abelian topological charges (NATCs), characterized by their noncommutative algebra, offer a framework for describing multigap topological phases beyond conventional Abelian invariants. While higher-order topological phases (HOTPs) host boundary states at corners or hinges, their characterization has largely relied on Abelian invariants such as winding and Chern numbers. Here, we propose a coupled-wire scheme of constructing non-Abelian HOTPs and analyze a non-Abelian second-order topological insulator as its minimal model. The resulting Hamiltonian supports hybridized corner modes, protected by parity-time-reversal plus sublattice symmetries and described by a topological vector that unites a non-Abelian quaternion charge with an Abelian winding number. Corner states emerge only when both invariants are nontrivial, whereas weak topological edge states of non-Abelian origins arise when the quaternion charge is nontrivial, enriching the bulk-edge-corner correspondence. The system further exhibits both non-Abelian and Abelian topological phase transitions, providing a unified platform that bridges these two distinct topological classes. Our work extends the understanding of HOTPs into non-Abelian regimes and suggests feasible experimental realizations in synthetic quantum systems, such as photonic or acoustic metamaterials.

Figures (7)

• Figure 1: Illustration of the 2D lattice model and its Brillouin zone. (a) shows the model realized by stacking 1D trimer chains. Along $x$ direction, each unit cell has three sublattices with onsite potentials $S_{A}$, $S_{B}$, $S_{C}$. Couplings between neighboring sites are indicated by red and green lines, corresponding to $J_{AB(BA)}$ and $J_{BC(CB)}$, respectively, where we take $J_{AB}=J_{BA}= i u$ and $J_{BC}=J_{CB}= i v$. Couplings between the same kind of sublattices are shown as black bonds, representing $J_{AA}$, $J_{BB}$ and $J_{CC}$. Along $y$ direction, coupling strengths alternate between $J_{1}$ and $J_{2}$ in purple and blue dashed lines. (b) shows the 2D Brillouin zone on a torus.
• Figure 2: Geometric illustrations of the Abelian winding number and non-Abelian quaternion charge. (a) shows the trajectory of Bloch vector $\mathbf{d}(k_{y})$ as $k_{y}$ varies from $-\pi$ to $\pi$. In the topological (trivial) phase with winding number $w=1$ ($w=0$), the trajectory of $\mathbf{d}(k_{y})$ in black solid line (pink dashed line) encircles once (does not encircle) the origin. (b)-(f) show the eigenframe rotation associated with each quaternion charge. The eigenstates $\ket{\psi_{1,2,3}(k_{x})}$ are marked by red, green, and blue dots at every $k_{x}$. The size of dots in each group increases progressively from $k_{x}=-\pi$ to $\pi$.
• Figure 3: Spectrum of 1D subsystems $H_x$ and $H_y$. All panels share the same color bar, which indicates the magnitude of IPR for each state. (a) shows the spectrum of $H_{y}$ in its topologically nontrivial region with $J_{1}=0.01$ and $J_{2}=2/5$. (b)--(e) show the spectrum and IPR of $H_{x}$. We set $\varphi=\pi$ for (b); $\varphi=3\pi/2$ for (c); $\varphi=0$ for (d); $\varphi=\pi/2$ For (e). Other parameters are set as $J_{AA}=-1$, $2J_{BB}=-u=v=1$, and $S_{A}=S_{B}=0$.
• Figure 4: Phase diagram of 2D non-Abelian HOTPs characterized by hybrid topological charge $\nu=(w,Q_x)$. Different SOTI phases are indicated by distinct colored regions, with their topological invariants shown therein. The system parameters are set as $S_{C}=2\cos{\varphi}$, $J_{CC}=\frac{1}{2}+\frac{1}{2}\sin{\varphi}$, $J_{2}=\frac{2}{5}$, $J_{AA}=-1$, $2J_{BB}=-u=v=1$, and $S_{A}=S_{B}=0$.
• Figure 5: Spectrum and corner states of the minimal non-Abelian HOTI model under OBC along both dimensions. (a)--(d) show the spectra of the system for different values of $\varphi$. The insets highlight the energy windows of corner states. The shared color bar indicates the IPR of each eigenstate. (e)--(h) show the probability distributions of corner states in the spectra (a)--(d). We set $J_{1}=0.01$, $J_{2}=0.4$, and $\varphi=\pi$ for (a), (e); $\varphi=3\pi/2$ for (b), (f); $\varphi=0$ for (c), (g); and $\varphi=\pi/2$ for (d), (h). Other parameters are equal to those of Fig. \ref{['fig: Phase_Diagram']}. The lattice size is $N=60$ and $M=40$.