Fermionic systems

• Hubbard dimer: charge fluctuations and superexchange
• Half-filled Hubbard ring: Mott correlations on a finite cluster
• Extended Hubbard ring: competing spin, charge, and bond correlations
• Heisenberg dimer in a constrained fermionic Hilbert space
• Flux-threaded ring: level flow and persistent current

The sequence moves from a solvable Hubbard bond to collective correlations on finite rings, competing order in the extended Hubbard model, a symmetry-constrained spin sector, and flux response in a complex Hermitian Hamiltonian.