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

TL;DR

This paper introduces a tensor-network approach that enables the calculation of excitonic spectra in super-moiré systems with over a billion lattice sites, surpassing traditional computational limits and revealing detailed excitonic features.

Contribution

The authors develop a novel tensor-network method for the real-space Bethe-Salpeter Hamiltonian, allowing large-scale excitonic spectrum computations in quasicrystal and super-moiré systems.

Findings

Successfully computed spectra for systems exceeding one billion sites.

Revealed exciton miniband formation and spatial confinement effects.

Demonstrated resolution of atomistic and mesoscopic structures in large systems.

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.