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We generalize the Su-Schrieffer-Heeger (SSH) model with the inclusion ofarbitrary long-range hopping amplitudes, providing a simple framework toinvestigate arbitrary adiabatic deformations that preserve the chiral symmetryupon the bulk energy bands with any arbitrary winding numbers. Using onlyelementary techniques of solving linear difference equations and applyingCauchy's integral formula, we obtain a mathematically rigorous and physicallytransparent proof of the bulk-boundary correspondence for the generalized SSHmodel. The multiplicity of robust zero-energy edge modes is shown to beidentical to the winding number. On the other hand, nonzero-energy edge modes,if any, are shown to be unstable under adiabatic deformations and not relatedto the topological invariant. Furthermore, under deformations of small spatialdisorder, the zero-energy edge modes remain robust.

An elementary rigorous proof of bulk-boundary correspondence in the generalized Su-Schrieffer-Heeger model