Cell Natural Orbitals: Quantifying Short-Range Entanglement in Quantum Materials

The quantum geometric tensor encodes how Bloch wavefunctions vary across the Brillouin zone, yet its physical consequences are most transparent in a local picture. A large quantum metric means that orbitals extend further, with larger position fluctuations of bound electrons in the lattice. This connects naturally to how entangled electrons are across different sites: topological obstructions in the band structure can enforce a minimum entanglement that cannot be removed by any smooth deformation. To quantify this, I will introduce Cell Natural Orbitals (CNOs), the eigenstates of the one-particle density matrix restricted to a single unit cell. Their eigenvalue spectrum measures directly how much orbital weight is shared between a cell and its neighbors through hybridization, distinguishing topological from trivial bands and identifying obstructed atomic limits. I will discuss applications to transition metal dichalcogenides and moiré heterostructures, where the local entanglement structure determines which correlated phases can emerge.