Universal Moiré-Model-Building Method without Fitting: Application to Twisted MoTe${}_2$ and WSe${}_2$

We develop a comprehensive method to construct analytical continuum models for moiré systems directly from first-principle calculations without any parameter fitting. The core idea of this method is to interpret the terms in the continuum model as a basis, allowing us to determine model parameters as coefficients of this basis through Gram-Schmidt orthogonalization. We apply our method to twisted MoTe${}_2$ and WSe${}_2$ with twist angles ranging from 2.13${}^{\circ }$ to 3.89${}^{\circ }$, producing continuum models that exhibit excellent agreement with both energy bands and wavefunctions obtained from first-principles calculations. We further propose a strategy to integrate out the higher-energy degrees of freedom to reduce the number of the parameters in the model without sacrificing the accuracy for low-energy bands. Our findings reveal that decreasing twist angles typically need an increasing number of harmonics in the moiré potentials to accurately replicate first-principles results. We provide parameter values for all derived continuum models, facilitating further robust many-body calculations. Our approach is general and applicable to any commensurate moiré materials accessible by first-principles calculations.