Flat bands without twists through the use of a one harmonic Moiré systems: topological nature of modes and electron–electron pairing in periodic uniaxial strained or crenellated graphene nanoribbons

Abstract Flat bands are known to produce highly correlated phases, leading to superconductivity and charge fractionalization. In two-dimensional systems, they have been extensively studied in magic-angle twisted bilayer graphene (MATBG). However, from both theoretical and experimental perspectives, MATBG remains a challenging system. Here, we present an alternative untwisted Moiré system to avoid such difficult twists. This system reproduces many of the interesting physical effects observed in Moiré systems, particularly flat bands and electron–electron pairing via a repulsive mechanism. The system consists of a graphene nanoribbon with periodic strain or corrugation, induced by a suitable substrate, such as a crenellated h-BN substrate. The strain is periodic with a single harmonic, but the key is to apply a spatial frequency slightly detuned from the condition that changes sign between neighboring sites. This produces a deformation that changes sign between neighbors in graphene while introducing a long-wavelength deformation visible only on each of the graphene’s bipartite lattices. This induces a spatially dependent effective mass, obtained using one harmonic. The system maps onto a Jackiw–Rebbi model, with flat-band modes identified as topological soliton modes. Electron–electron interactions are included in the system using the Hubbard Hamiltonian. The main result is the emergence of an effective attraction between electrons, accompanied by spin polarization coupled to the electron pseudospin. These observations align with the Kohn–Luttinger theory of superconductivity in other bipartite lattices, such as high-Tc cuprates. Since the Jackiw–Rebbi model exhibits charge fractionalization, similar phases are also expected to appear, as observed in MATBG.