Photonic topological pump between chiral disclination states

A topological pump, a quantum pump protected by the bulk topology, enables the robust quantized transport of particles (or waves) when the system parameters are varied in a cyclic process. In previous studies, topological pumps were studied in undistorted systems. Here, we show topological pumps can happen between disclination states with different chiralities in a distorted lattice. Based on a generalized understanding of the charge pumping process, we explain the topological disclination pump by tracing the motion of Wannier centers in each unit cell. In addition, by constructing two disclination structures and introducing a symmetry-breaking perturbation, we achieve a topological pumping between different disclination cores. Furthermore, we expect to observe our predictions in femtosecond-laser-written waveguide arrays with a fine-tuned geometry and refractive index. Our result opens a route to study the topological pump in distorted systems and enables the quantized transport of photons between real-space topological defects.

Figure 1

The formation of disclinations and eigenstates. (a) The original undistorted lattice. The shadow area is removed to construct a ${\mathrm{C}}_4$ symmetric disclination structure. The parameters in $\mathcal{H}$ are shown in the lower panel. The orange (blue) dots represent $A$ ($B$) sublattices. The red (purple) lines represent the intercell (intracell) coupling. (b) A ${\mathrm{C}}_4$ symmetric disclination structure. (c) The disclination structure with removing some boundary sites. (d) The eigenstates of (b) where there are bulk states (black dots), edge states (blue dots), corner states (green triangles), and disclination states (red dots) whose field distributions are mainly distributed in the gray, blue, green, and red unit cells as shown in (b). (e) The eigenstates of the disclination structure with removing some boundary sites. There are no zero-energy corner states.

Figure 2

Parameter evolutions for the topological pump. (a) The evolution of $\mu$ (black line), $v$ (blue line), and $w$ (red line) as functions of $t$ in a period. (b) Schematic of the evolution of the Hamiltonian $\mathcal{H}$ in the parameter space. For the topological pump, the trajectory encircles the gapless phase transition point (orange line) while for trivial evolution, the trajectory does not. (c) The motion of three Wannier centers (black dots) during the pumping process (orange lines). (d) The motion of three Wannier centers (black dots) during the trivial evolution process (gray lines).

Figure 3

The local topological disclination pump. (a) Implementation based on the photonic waveguide array. The red and blue color on the waveguides represents the decrease and increase of the refractive index. (b) The spectral flow of the topological pump within a single disclination core. The disclination states (red line) with different chiralities (labeled by $+$ and $-$) intersect each other during the pumping process. (c)–(g) The field distributions of the initial, intermediate, and final states labeled by 1, 2, 3, 4, 5 in (a), respectively.

Figure 4

Nonlocal topological disclination pumping. (a) Implementation based on the photonic waveguide array. The red and blue color on the waveguides represents the decrease and increase of the effective refractive index. (b) The lattice structure at the cross section labeled by 1 in (a). There are two disclination cores with the same parameters as in Fig. 1 with only four sites (sites in green disks) perturbed by adding an on-site energy $0.5\mu$. (c) Spectral flow of (a) with the same parameters as in Fig. 3.