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

TL;DR

This study uses exact diagonalization on a $3\times 4$ cylinder to map how hole binding and competing spin–charge orders emerge in the simple ($V=0$) and extended Hubbard models. By tuning on-site $U$ and nearest-neighbor $V$, the work identifies distinct binding mechanisms: magnetically mediated two-hole pairing at $V=0$, attraction-driven phase separation with magnetic quenching for $V<0$, and CDW-dominated backgrounds constraining pairing for $V>0$, with stronger effects at $U=10$. The results are corroborated by analyses of binding energies $E_{B2}$, $E_{B3}$, $E_{B4}$, structure factors $S_L(\pi,\pi)$ and $S_D(\pi,\pi)$, and real-space maps of $L_{ij}$ and $D_{ij}$, revealing a coherent picture where nonlocal interactions reshape the pairing landscape in minimal correlated-electron models. The findings bridge finite-size ED insights with themes relevant to larger systems and potential experimental realizations in engineered quantum materials.

Abstract

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.

Figures (16)

• Figure 1: The excitation gap $\Delta E$ as a function of $U$ for the half-filled ($N_h=0$), one-hole ($N_h=1$), two-hole ($N_h=2$), three-hole ($N_h=3$) and four-hole ($N_h=4$) doped systems of the $3\times 4$ cylindrical lattice.
• Figure 2: Single-particle molecular orbital (MO) energy spectra of the non-interacting ($U=0$) half-filled ($N_h=0$) Hubbard model on the $3\times 4$ cylindrical lattice. Degeneracy at the Fermi level for the $N_h=0$ and $N_h=1$ cases yields a vanishing gap.
• Figure 3: Two-, three- and four-hole binding energies as a function of $U$ in the ground state of the Hubbard model with $N_h=2$, 3 and 4 holes, respectively, on $3\times 4$ cylindrical lattice. The inset compares these results with the corresponding lattice with open boundary conditions (OBC).
• Figure 4: Comparison of (a) nearest-neighbor spin correlation ($L_{1,2}$) and (b) next nearest-neighbor spin correlation ($L_{1,3}$), as a function of number of holes $N_h$ at $U=0$ and several positive $U$ values on $3\times 4$ cylindrical lattice.
• Figure 5: Comparison of (a) nearest-neighbor charge correlation ($D_{1,2}$) and (b) next nearest-neighbor charge correlation ($D_{1,3}$), as a function of number of holes $N_h$ at $U=0$ and several negative $U$ values on $3\times 4$ cylindrical lattice.