Two-dimensional non-Abelian topological insulators and the corresponding edge/corner states from an eigenvector frame rotation perspective

We propose the concept of two-dimensional (2D) non-Abelian topological insulators which can explain the energy distributions of the edge states and corner states in systems with parity-time symmetry. From the viewpoint of non-Abelian band topology, we establish the constraints on the 2D Zak phase and polarization. We demonstrate that the corner states in some 2D systems can be explained as the boundary mode of the one-dimensional edge states arising from the multiband non-Abelian topology of the system. We also propose the use of the off-diagonal Berry phase as complementary information to assist the prediction of edge states in non-Abelian topological insulators. Our work provides an alternative approach to study edge and corner modes and this idea can be extended to three-dimensional systems.

Figure 1

Non-Abelian topological insulator. (a) The eigenstate rotations in a 1D non-Abelian topological insulator. (b) A 1D non-Abelian topological insulator can be viewed as a subsystem which is a closed loop encircling nodal lines (degeneracies) in a higher dimension parameter space. (c) The 1D band structure of a flat-band model. (d) The eigenstate pattern in a 2D non-Abelian topological insulator. (e) A 2D non-Abelian topological insulator can be viewed as a torus surface encircling nodal lines in 3D. (f) The 2D band structure of a flat-band model. (g) The eigenstate pattern which does not satisfy Bloch condition. (h) Forbidden nodal line configurations impose constraints on the 2D polarizations. (i) Two paths in a 2D BZ. (j) The eigenstate pattern with singularities. (k) An admissible nodal line configuration with two nodal lines penetrating the torus surface. (l) The band structure with degeneracies.

Figure 2

Three-band 2D non-Abelian topological insulator. (a) Tight-binding model. (b) The energy spectrum of a finite sample with 10 × 10 unit cells. The gray, red, and blue states are bulk, vertical, and horizontal edge states, respectively. (c) The band structure of a vertical ribbon. (d) The band structure of a horizontal ribbon. (e)–(g) The representative field patterns of vertical (e) and horizontal (f), (g) edge states.

Figure 3

Four-band 2D non-Abelian topological insulator. (a) The tight-binding model. (b) 2D band structure. (c), (d) The band structures of a vertical ribbon and a horizontal ribbon, respectively. (e) The energy spectrum of a $10\times 10$ lattice. (f)–(h) Examples of the strength field patterns of vertical edge, horizontal edge, and corner states, respectively. (i) The orthographical projections of the eigenstate rotations along the path $k_y=0, k_x:-\pi \to \pi$. (j) The orthographical projections of the eigenstate rotations along the path $k_x=0, k_y:-\pi \to \pi$. (k) The geometrical meaning of the Euler connection. The shaded area is the projected area scanned by the $|{\phi }^m\left(\mathit{k}\right)\rangle$ (or $|{\phi }^n\left(\mathit{k}\right)\rangle$) on the $m\text{−}n$ plane in the $\mathrm{\Delta }\mathit{k}$. (l) The OBP curves ${\gamma }_x^{mn}\left(k_y\right)$. (m) The OBP curves ${\gamma }_y^{mn}\left(k_x\right)$.

Figure 4

A model with degeneracy points. (a) The tight-binding model with the diagonal and antidiagonal hopping. (b) The 2D band structure. There are two degeneracy points between the second and third bands. (c) The corresponding nodal line configuration viewed from higher dimensions. (d) Top panel: The non-Abelian charges $q_y\left(k_x\right)$ in different subdomains of the 2D BZ. Middle panel: The Euler phase curves ${\gamma }_y^{mn}\left(k_x\right)$. Bottom panel: The band structure of a horizontal ribbon (see inset). (e) Left panel: The non-Abelian charges $q_x\left(k_y\right)$. Middle panel: The Euler phase curves ${\gamma }_y^{mn}\left(k_x\right)$. Right panel: The band structure of a vertical ribbon (see inset). (f) The energy spectrum of the $10\times 10$ lattice. (g),(h) Examples of the vertical edge and horizontal edge states, respectively.