Edge modes and topological phases in a generalized Su-Schrieffer-Heeger ladder system

Motivated by the realization of topological edge states in Su-Schrieffer-Heeger (SSH) chains in cold atomic systems, we study a ladder model consisting of two coupled SSH chains. Such a system can serve as a testbed for study and detection of edge modes analogous to the bound Majorana states in the Kitaev wire. The SSH ladder Hamiltonian belongs to the BDI symmetry class and is characterized by a Z-valued topological invariant. When the two chains have identical hopping but are off-set by one lattice site, the topological phase of the SSH ladder is characterized by the existence of Dirac zero energy modes at the edge. These modes are similar to the Majorana modes of the Kitaev chain with respect to the spatial profile of associated wavefunctions. For a more general model of coupled SSH chains having four distinct hopping amplitudes, we show that a more complex phase diagram is obtained as the system can support another kind of edge states. Through numerical and analytical methods, we study the distinct topological phases hosting these modes and their robustness with respect to the strength of the inter-chain coupling. Relevant for experimental systems, we additionally consider the effects of finite size on the edge-mode structure and the phase diagram.