Topological phases of generalized Su-Schrieffer-Heeger models

We study the extended Su-Schrieffer-Heeger model with both nearest-neighbor and next-nearest-neighbor hopping strengths cyclically modulated and find the family of the model system exhibiting topologically nontrivial phases, which can be characterized by a nonzero Chern number defined in a two-dimensional space spanned by the momentum and a modulation parameter. It is interesting that the model has a similar phase diagram to that of the well-known Haldane model. We propose to use photonic crystal systems as the ideal systems to fabricate our model systems and probe their topological properties. Some other models with modulated on-site potentials are also found to exhibit similar phase diagrams, and their connection to the Haldane model is revealed.

Figure 1

Energy spectrum for the extended SSH model with 100 sites and $\delta =0.5$ under OBCs. (a) $t_A=t_B=0$. (b) $t_A=t_B=0.3$. (c) $t_A=-t_B=0.2$. (d) $t_A=0.5,t_B=0.1$.

Figure 2

Energy spectrum for the extended SSH model with 100 sites, $\delta =0.5$, and $h=0.2$ under OBCs. (a), (b), and (c) correspond to $\varphi =-\pi /4$, 0, and $\pi /4$, respectively. Insets show the wave functions of edge modes.

Figure 3

Phase diagram for the extended SSH model with the NNN hopping amplitude given by Eq. (3). The number $C$ indicates the Chern number of the lower band.

Figure 4

Energy spectrum of the extended SSH model with 100 sites and $\delta =0.5$ under OBCs. (a) $g_{+}=0.3,h=0.15,g_{-}/h=0.5,\varphi =0.5\pi$. Each of (b)–(e) has only one parameter different from (a). (b) $h=0.1$. (c) $g_{+}=0.4$. (d) $g_{-}/h=1.5$. (e) $\varphi =0$. Insets show the wave functions of edge modes.

Figure 5

(a) Schematic diagram for the arrangement of waveguides for a given $\theta$. (b) Along the propagation direction, $\theta$ can be varied along the length of the waveguide by smoothly changing the spacing between successive waveguides. (c) Examples for the geometrical arrangement of waveguides for several $\theta$. Red circles stand for site A and green ones stand for site B in the model.