Self-consistent tensor network method for correlated super-moiré matter beyond one billion sites
Yitao Sun, Marcel Niedermeier, Tiago V. C. Antão, Adolfo O. Fumega, Jose L. Lado
TL;DR
This work tackles the formidable challenge of simulating correlated states in super-moiré materials with billions of sites by developing a self-consistent tensor-network framework. It maps the real-space mean-field problem to a linearly scaling pseudo-spin MPO representation and combines a Chebyshev kernel polynomial method with quantics tensor cross interpolation to compute self-consistent correlators directly on MPOs. The approach enables self-consistent solutions, symmetry-broken states, and spectral functions for systems exceeding a billion sites, demonstrated in both 1D and 2D super-moiré geometries with spatially modulated hoppings and interactions, including domain walls. By avoiding explicit storage of the full single-particle Hamiltonian and exploiting structured modulations, the method achieves dramatic scalability gains and opens the door to real-space dynamical extensions and real-space DFT treatments of ultra-large correlated quantum matter.
Abstract
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, that 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 exponentially large 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.