Binding of holes and competing spin-charge order in simple and extended Hubbard model on cylindrical lattice: An exact diagonalization study

We investigate the binding of holes and the emergence of competing spin-charge order in the simple and extended Hubbard model using exact diagonalization on the 3x4 cylindrical lattice. For the simple Hubbard model (V=0), we find weakly bound hole pairing mediated by magnetic correlations at intermediate repulsive U, without any evidence of phase separation. Introducing nearest-neighbor interaction V reveals a rich phase diagram: attractive V drives multi-hole clustering and phase separation with localized magnetic quenching, while repulsive V stabilizes charge-density-wave (CDW) order that coexists with bound hole pairs within a modulated magnetic background. At strong coupling (U=10), the competition sharpens, with attractive V overcoming on-site repulsion to form magnetically quenched clusters and repulsive V producing robust CDW order that constrains pairing. Real-space analysis of spin and charge correlations provides microscopic evidence of distinct binding mechanisms – phase separation versus correlation-mediated pairing – depending on the sign and strength of intersite interaction V . Our results establish a comprehensive picture of how nonlocal Coulomb interactions reshape the landscape of hole-binding and collective order in correlated electron systems.

💡 Research Summary

In this work the authors perform an exact‑diagonalization (ED) study of the Hubbard model and its extended version on a 3 × 4 cylindrical lattice (12 sites) with periodic boundary conditions along the x‑direction and open boundaries along y. The Hamiltonian includes nearest‑neighbor hopping t, on‑site repulsion U and a nearest‑neighbor interaction V. By varying U (from attractive to strongly repulsive) and V (both attractive and repulsive) they compute ground‑state energies for hole dopings Nh = 0–4 and evaluate binding energies for two, three and four holes (EB2, EB3, EB4). A negative EB2 signals effective attraction between two holes, while negative EB3 and EB4 together indicate phase separation.

For the simple Hubbard model (V = 0) the authors find that at intermediate repulsive U (≈ 4 t) the two‑hole binding energy is weakly negative (≈ −0.1 t), suggesting a modest correlation‑induced pairing tendency. The binding is strongest for two holes; three‑ and four‑hole binding energies remain positive for all U, indicating that larger clusters do not form and that the system does not phase‑separate on this finite lattice. Real‑space spin‑spin (Lij) and charge‑charge (Dij) correlators reveal a magnetic‑polaron picture: a single hole drags a cloud of disturbed spins, while a second hole can retrace the path and restore the spin background, thereby gaining kinetic energy.

Introducing a nearest‑neighbor interaction V dramatically reshapes the landscape. For attractive V (V < 0) the two‑hole binding energy becomes strongly negative (e.g., EB2 ≈ −2.5 t at U = 10, V = −2), and EB3 and EB4 also turn negative, signalling the formation of multi‑hole clusters and macroscopic phase separation. In this regime the charge‑charge correlator shows a pronounced peak inside the cluster, while the spin correlator is strongly suppressed, indicating a magnetically quenched region where local magnetic order is destroyed by the dense hole aggregation.

Conversely, repulsive V (V > 0) suppresses hole pairing. EB2 remains only slightly negative (≈ −0.05 t) and EB3, EB4 stay positive, so phase separation is absent. Instead, a charge‑density‑wave (CDW) pattern emerges: Dij exhibits a periodic modulation of charge density across the lattice, and Lij retains a modulated antiferromagnetic background. The CDW competes with hole binding, limiting the mobility of hole pairs. The competition is most pronounced at strong coupling (U = 10), where the sign of V determines whether the system falls into a phase‑separated, magnetically quenched state (V < 0) or a CDW‑stabilized, weakly paired state (V > 0).

The authors also compare periodic and open boundary conditions. While quantitative values of the binding energies shift, the qualitative trends—weak pairing for V = 0, phase separation for V < 0, and CDW for V > 0—are robust. This demonstrates that even on a small cluster the non‑local interaction V is the decisive factor governing whether holes bind via magnetic correlations, cluster and drive phase separation, or coexist with charge order.

In summary, the paper establishes that (i) the simple Hubbard model on a 12‑site cylinder supports only modest two‑hole pairing at intermediate U, (ii) an attractive nearest‑neighbor interaction can overcome on‑site repulsion to produce multi‑hole clusters and phase separation with quenched magnetism, and (iii) a repulsive V stabilizes charge‑density‑wave order that coexists with weakly bound hole pairs within a modulated magnetic background. These exact results provide microscopic insight into how non‑local Coulomb terms reshape spin‑charge competition, offering a valuable benchmark for larger‑scale numerical methods and for experiments on ultracold atoms in optical lattices that emulate extended Hubbard physics.