Tensor-network methodology for super-moir\'e excitons beyond one billion sites

Abstract

Computing excitonic spectra in quasicrystal and super-moir\'e systems constitutes a formidable challenge due to the exceptional size of the excitonic Hilbert space. Here, we demonstrate a tensor-network method for the real-space Bethe-Salpeter Hamiltonian, allowing us to access the spectra of an excitonic $10^{18}$-dimensional Hamiltonian, and enabling the direct computation of bound-exciton spectral functions for systems exceeding one billion lattice sites, several orders of magnitude beyond the capabilities of conventional approaches. Our method combines a tensor-network encoding of the real-space Bethe-Salpeter Hamiltonian with a Chebyshev tensor network algorithm. This strategy bypasses explicit storage of the Hamiltonian while preserving full real-space resolution across widely different length scales. We demonstrate our methodology for one- and two-dimensional super-moir\'e systems, achieving the simultaneous resolution of atomistic and mesoscopic structures in the excitonic spectra in billion-size systems, showing exciton miniband formation and moir\'e-induced spatial confinement. Our results establish a real-space methodology enabling the simulation of excitonic physics in large-scale quasicrystal and super-moir\'e quantum matter.