Higher-order topological phases emerging from Su-Schrieffer-Heeger stacking

In this work, we develop a systematic approach of constructing and classifying the model Hamiltonians for two-dimensional (2D) higher-order topological phase with corner zero-energy states (CZESs). Our approach is based on the direct construction of an analytical solution of the CZESs in a series of 2D systems stacking the 1D extended Su-Schrieffer-Heeger (SSH) model, two copies of the two-band SSH model, along with two orthogonal directions. Fascinatingly, our approach not only gives the celebrated Benalcazar-Bernevig-Hughes and 2D SSH models but also reveals a novel model and we name it 2D crossed SSH model. Although these three models exhibit completely different bulk topology, we find that the CZESs can be universally characterized by edge winding number for 1D edge states, attributing to their unified Hamiltonian construction form and edge topology. Remarkably, our principle of constructing CZESs can be readily generalized to 3D and superconducting systems. Our work sheds new light on the theoretical understanding of the higher-order topological phases and paves the way to looking for higher-order topological insulators and superconductors.