Self-consistent tensor network method for correlated super-moiré matter beyond one billion sites

Moiré and super-moiré materials provide exceptional platforms to engineer exotic correlated quantum matter. The vast number of sites required to model moiré systems in real space remains a formidable challenge due to the immense computational resources required. Super-moiré materials push this requirement to the limit, where millions or even billions of sites need to be considered, a requirement beyond the capabilities of conventional methods for interacting systems. Here, we establish a methodology that allows solving correlated states in systems reaching a billion sites, which exploits tensor network representations of real-space Hamiltonians and self-consistent real-space mean-field equations. Our method combines a tensor network kernel polynomial method with quantics tensor cross-interpolation algorithm, enabling us to solve ultralarge models, including those whose single-particle Hamiltonian is too large to be stored explicitly. We demonstrate our methodology with super-moiré systems featuring spatially modulated hoppings, many-body interactions, and domain walls, showing that it allows access to self-consistent symmetry-broken states and spectral functions of real-space models reaching a billion sites. Our methodology provides a strategy to solve exceptionally large interacting problems, providing a widely applicable strategy to compute correlated super-moiré quantum matter.