Interacting ground states of moir\'e ladders

Abstract

Moir\'e materials have emerged as a rich platform for exploring strong correlation effects in low dimensions, with twisted bilayer graphene (TBG) as a paradigmatic example. To distill the essential ingredients driving moir\'e-induced phases, a simplified one-dimensional analog -- a two-leg ladder with spatially modulated interleg hopping and a uniform magnetic flux -- was recently introduced. This model, which we refer to as the moir\'e ladder, features a nearly flat lowest-energy band in a suitable parameter regime, capturing the band-flattening mechanism of TBG. We investigate the ground-state phase diagram of the moir\'e ladder using a combination of bosonization and density matrix renormalization group (DMRG) techniques, and systematically disentangle the respective roles of the flux and the hopping modulation. At half filling, previous numerical work identified a metal-insulator transition at finite interaction strength and an unexpected ferromagnetic ground state. Revisiting this, we show that the metal-insulator transition can be understood perturbatively within bosonization, governed by the number of Fermi points. In contrast, the ferromagnetic correlations are nonperturbative and require both flux and spatial modulation -- neither alone is sufficient. We extend our analysis to other fillings: one-quarter, three-quarters, slightly above half filling (half filling plus two electrons), and slightly below half filling (half filling minus two electrons). At moderate interactions, we observe ferromagnetism below half filling and antiferromagnetism above; at stronger interactions, ferromagnetism dominates across all studied fillings. Crucially, the analysis demonstrates that periodic interleg hopping alone does not engender new correlated phases; the magnetic flux is essential for the observed unconventional behavior.