  Topological insulators (TI) are bulk insulators that possess robust chiral
conducting states along their interfaces with normal insulators. A tremendous
research effort has recently been devoted to TI-based heterostructures, in
which conventional proximity effects give rise to many exotic physical
phenomena. Here we establish the potential existence of "topological proximity
effects" at the interface of a topological graphene nanoribbon (GNR) and a
normal GNR. Specifically, we show that the location of the topological edge
states exhibits versatile tunability as a function of the interface
orientation, as well as the strengths of the interface coupling and spin-orbit
coupling in the normal GNR. For zigzag and bearded GNRs, the topological edge
state can be tuned to be either at the interface or outer edge of the normal
ribbon. For armchair GNR, the potential location of the topological edge state
can be further enriched to be at the edge of or within the normal ribbon, at
the interface, or diving into the topological GNR. We also discuss potential
experimental realization of the predicted topological proximity effects, which
may pave the way for integrating the salient functionality of TI and graphene
in future device applications.
