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We investigate the topological interface modes of surface plasmon polaritons in a multilayer system composed of graphene waveguide arrays. The topological interface modes emerge when two topologically distinct graphene multilayer arrays are connected. In such multilayer system, the non-trivial topological interface modes and trivial modes coexist. By tuning the configuration of the graphene multilayer arrays, the associated non-trivial interface modes present robust against structural disorder. The total number of topological modes is related to that of graphene layers in a unit cell of the graphene multilayer array. The results provide a new paradigm for topologically protected plasmonics in the graphene multilayer arrays. The study suggests a promising approach to realize light transport and optical switching on a deep-subwavelength scale. [ABSTRACT FROM AUTHOR]

• We study theoretically and numerically both commensurate and incommensurate AAH graphene lattices and realize the topological edge modes of SPPs and localization transition from extended to all-localized wavefunctions, respectively. • The graphene systems can enable flexible and efficient modulation for constructing AAH lattice, thus circumventing the drawbacks of limited modulation depths in all dielectric AAH waveguide systems. • The topological edge states and localization transitions for graphene SPPs may be utilized for developing plasmonics devices operating in deep subwavelength scale on fully flat and compact platforms. By using rational and irrational on-site modulation in graphene waveguide arrays, we construct both commensurate and incommensurate Aubry-André-Harper (AAH) models and achieve the topological edge states and localization transition for graphene surface plasmon polaritons (SPPs). We show that when the commensurate AAH graphene array is truncated, topological edge states emerge, which are immune to the random perturbation of modulation depth in each individual graphene sheet. While for an incommensurate AAH array, localization transition for SPPs modes have also been achieved when the modulation depth reaches a critical value. Our work points out that graphene waveguide array provides a promising platform for robust light transport and compact localization within a fully deep subwavelength scale. [ABSTRACT FROM AUTHOR]