Van der Waals-coupled electronic states in incommensurate double-walled carbon nanotubes

Non-commensurate two-dimensional materials such as a twisted graphene bilayer or graphene on boron nitride, consisting of components that have no finite common unit cell, exhibit emerging moiré physics such as novel Van Hove singularities, Fermi velocity renormalization, mini Dirac points and Hofstadter butterflies. Here we use double-walled carbon nanotubes as a model system for probing moiré physics in incommensurate one-dimensional systems, by combining structural and optical characterizations. We show that electron wavefunctions between incommensurate inner- and outer-wall nanotubes can hybridize strongly, contrary to the conventional wisdom of negligible electron hybridization due to destructive interference. The chirality-dependent inter-tube electronic coupling is described by one-dimensional zone folding of the electronic structure of twisted-and-stretched graphene bilayers. Our results demonstrate that incommensurate van der Waals interactions can be important for engineering the electronic structure and optical properties of one-dimensional materials.