Topological properties of non-isotropic two-dimensional SSH model

<sec>The one-dimensional (1D) Su-Schrieffer-Heeger (SSH) chain is a model that has been widely studied in the field of topological physics. The two-dimensional (2D) SSH model is a 2D extension of the 1D SSH chain and has many unique physical properties. It is a higher-order topological insulator (HOTI), in which corner states with bound states in the continuum (BIC) properties will arise between the second energy band and the third energy band. There are two different topological phases in the isotropic 2D SSH model, and a topological phase transition will happen when the intracell coupling strength is equal to the intercell coupling strength.</sec><sec>In this paper, we first break the isotropy of the isotropic 2D SSH model, defining the ratio of the <i>x</i>-directional coupling strength to the <i>y</i>-directional coupling strength as <i>α</i> and the ratio of the intercell coupling strength to the intracell coupling strength as <i>β</i>, which represent the strength of the topological property and anisotropy respectively. We use <i>α</i> and <i>β</i> to calibrate all possible models, classify them as three different types of phases, and draw their phase diagrams.Then we argue when the energy gap between the second energy band and the third energy band emerges over the entire Brillouin zone.</sec><sec>Meanwhile, we use a method to calculate the spatial distribution of polarization when the model is half-filled, and it is shown that there is 1/2 polarization localized at the edges in the direction with larger intracell coupling, but no edge polarization in the other direction. The edge polarization excites the edge dipole moment, giving rise to a topological edge state in the energy gap. At the same time, when the model has an entire open boundary, the dipole moment directs the charge to accumulate on the corners, which can be observed from the local charge density distribution. This type of fractional charge is a filling anomaly and formed spontaneously by the lattice to maintain electrical neutrality and rotational symmetry simultaneously. This fractional charge induces the aforementioned corner state. And by its nature of filling anomaly, this corner state is better localized and robust. It will not couple with the bulk state as long as the rotational symmetry or chirality of the model is not broken.</sec><sec>Finally, we construct an acoustic resonant cavity model: a rectangular shaped resonant cavity is used to simulate individual lattice points and the coupling strength between the lattice points is controlled by varying the diameter of the conduit between the resonant cavities. According to the Comsol calculation results, we can see that the topological properties of the anisotropic two-dimensional SSH model are well simulated by this model.</sec>