Effective tight-binding models in optical moiré potentials

A twist between two periodic systems offers the possibility to drastically control quantum transport properties. At discrete sets of commensurate twisting angles, a moir\'e periodic pattern is formed, and the low-energy single-particle spectrum forms distinct bands and gaps. Here, we consider twisted systems at a commensurate angle made of two continuous-space periodic potentials. We devise a general approach, based on a hierarchy of energy scales, to derive effective tight-binding models that capture the energy spectrum, focusing on the case of square lattices. Localized states in the moir\'e unit cell are coupled by strong intracell and weak intercell tunnelings, generating a series of energy bands and subbands that correspond to distinct, periodic lattices. For sufficiently deep potentials, we find excellent agreement between the spectrum of the continuous-space system and the tight-binding approximation thus constructed. Our approach allows for the interpretation of moir\'e band structures and the extraction of parameters for the effective tight-binding models. Applications to the physics of many-body systems and extensions to other twisted models are also discussed.