General bounded corner states in the two-dimensional Su-Schrieffer-Heeger model with intracellular next-nearest-neighbor hopping

We investigate corner states in a photonic two-dimensional (2D) Su-Schrieffer-Heeger (SSH) model on a square lattice with zero gauge flux. By considering intracelluar next-nearest-neighbor (NNN) hoppings, we discover a broad class of corner states in the 2D SSH model and show that they are robust against certain fabrication disorders. Moreover, these corner states are located around the corners but not at the corner points. We analytically identify that these corner states are induced by the intracelluar NNN hoppings (long-range interactions) and split off from the edge-state bands. Thus, we refer to them as general bounded corner states. Our paper shows a simple way to induce unique corner states by the long-range interactions and offers opportunities for designing novel photonic devices.

Figure 1

(a) Schematic of a photonic 2D SSH model on a square lattice. There are four modes ($A\text{–}D$) in one unit cell with lattice constant $2a$, and the amplitudes of intracelluar and intercelluar nearest-neighbor hoppings are $\gamma$ and $\lambda$, respectively. The red oblique lines represent intercelluar NNN hoppings in unit cells with strength $J$. The possible realistic physical systems to implement one unit cell of the photonic 2D SSH model: (b) network of coupled superconducting transmission line resonators [74, 75, 76] and (c) 2D lattice of nanophotonic silicon ring resonators [66].

Figure 2

(a) Energy spectrum for a 2D SSH model with a $20\times 20$ array of unit cells with $\gamma =\lambda /5$ and $J=0$. The orange (gray) bands denote the the topological ESs of the lattice, the bulk states (BSs) are depicted by yellow (light gray) bands and the four degenerate zero-energy corner states (ZECSs) appear at zero energy with a red (dark gray) point. (b) The local density of states (LDOS) with where the ZECS for mode $A$ at lattice site (1,1), the ES for mode $A$ at lattice site (1,11), and the BS for mode $A$ at lattice site (11,11). The field profiles of the corner states without disorder in (c) and with disorder $\epsilon \in [-\lambda /100,\lambda /100]$ in (d).

Figure 3

(a) Energy spectrum for a 2D SSH model with a $20\times 20$ array of unit cells with $\gamma =\lambda /5$ and $J=\lambda /2$. (b) Corresponding LDOS with $\kappa =\lambda /20$. The general bounded corner states [CS1–CS4, LDOS for mode $B$ at lattice site (1,1)] appear in the band gaps between the edge and bulk bands. The field profiles of the general bounded corner states CS1–CS4 in (c)–(f) with disorder $\epsilon \in [-\lambda /100,\lambda /100]$.

Figure 4

(a) Energy spectrum of a 2D SSH model with varying intercellular NNN hopping $J/\lambda$ with $\gamma =\lambda /5$. (b) Illustration of coupling between modes around the corners forming general bounded corner states with $\gamma =0$. (c) Energy spectrum of a 2D SSH model with varying intercellular NNN hopping $J/\lambda$ with $\gamma =\lambda /20$. (d) Enlarged section of (c) showing the emergence of more corner states from the bands of edge states as $J$ increases where the colored (dotted orange, short dashed blue, dash-dotted green, and dashed red) curves show the analytical result for the energies of corner states obtained with modes around the corners highlighted in the same color in (b).

Figure 5

(a) Energy spectrum for a 2D SSH model with a $20\times 20$ array of unit cells with $\gamma =\lambda /20$ and $J=0.8\lambda$. (b) Enlarged section of (a) showing four general bounded corner states: CS1–CS4. (c) LDOS of the general bounded corner states CS1–CS4 with $\kappa =\lambda /20$ for mode $B$ at lattice sites (1,1), (2,2), (3,3), and (4,4) respectively. (d)–(g) Corresponding field profiles of the corner states CS1–CS4 with disorder $\epsilon \in [-\lambda /100,\lambda /100]$.