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arXiv:cond-mat/9701217 · Physical Review Letters

Finite-Size Studies on the SO(5) Symmetry of the Hubbard Model

Stefan Meixner, Werner R. Hanke, Eugene Demler, Shoucheng Zhang

Published Dec 15, 1997• 43 citations
arXivHTMLPDF

Abstract

We present numerical evidence for the approximate SO(5) symmetry of the Hubbard model on a 10-site cluster. Various dynamic correlation functions involving the $\ensuremath{\pi}$ operators, the generators of the SO(5) algebra, are studied using exact diagonalization, and are found to possess sharp collective peaks. Our numerical results also lend support to the interpretation of the recent resonant neutron scattering peaks in the YBCO superconductors in terms of the Goldstone modes of the spontaneously broken SO(5) symmetry. Perturbations such as longer-ranged hoppings and interactions do not suppress the $\ensuremath{\pi}$ resonance in interesting parameter regimes.

DOI
10.1103/physrevlett.79.4902

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We present numerical evidence for the approximate SO(5) symmetry of the Hubbard model on a 10-site cluster. Various dynamic correlation functions involving the \ensuremathπ\ensuremath{\pi}\ensuremathπ operators, the generators of the SO(5) algebra, are studied using exact diagonalization, and are found to possess sharp collective peaks. Our numerical results also lend support to the interpretation of the recent resonant neutron scattering peaks in the YBCO superconductors in terms of the Goldstone modes of the spontaneously broken SO(5) symmetry. Perturbations such as longer-ranged hoppings and interactions do not suppress the \ensuremathπ\ensuremath{\pi}\ensuremathπ resonance in interesting parameter regimes.
In this work, we present numerical results which support SO(5) symmetry as a concept unifying superconductivity and antiferromagnetism in the high-temperature superconductors. Using exact cluster diagonalization, we verify the recently proposed SO(5) multiplet structures for a widely used microscopic model, the t-J model. Our results show that the d-wave superconducting ground states away from half-filling are obtained from the higher spin states at half-filling through SO(5) rotations, that the dominant low energy resonance in the dynamical spin correlation function corresponds to a `pi-resonance' in the entire low doping region, and that the SO(5) symmetry is robust against inclusion of longer ranged Coulomb repulsion and next-nearest neighbor hopping.
In this work, we present numerical results which support SO(5) symmetry as a concept unifying superconductivity and antiferromagnetism in the high-temperature superconductors. Using exact cluster diagonalization, we verify that low-energy states of the $t\ensuremath{-}J$ model, a widely used microscopic model for the high-Tc{T}_{c}Tc​ cuprates, form SO(5) symmetry multiplets. Our results show that the ddd-wave superconducting ground states away from half-filling are obtained from the higher-spin states at half-filling through SO(5) rotations. The SO(5) symmetry is found to be robust against the inclusion of longer-ranged Coulomb repulsion and next-nearest-neighbor hopping.
In this paper, we present analytical and numerical calculations of the \ensuremath{\pi} resonance in the $t\ensuremath{-}J$ model. We show in detail how the $\ensuremath{\pi}$ resonance in the particle-particle channel couples to the dynamical spin correlation function in a superconducting state. The contribution of the \ensuremath{\pi} resonance to the spin excitation spectrum can be estimated from general model-independent sum rules, and it agrees with our detailed calculations. The results are in overall agreement with the exact diagonalization studies of the $t\ensuremath{-}J$ model. Earlier calculations predicted the correct doping dependence of the neutron resonance peak in the YBa2Cu3O6+x{\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6+x}YBa2​Cu3​O6+x​ superconductor, and in this paper detailed energy and momentum dependence of the spin correlation function is presented. The microscopic equations of motion obtained within current formalism agree with that of the SO(5) nonlinear \ensuremath{\sigma} model, where the \ensuremath{\pi} resonance is interpreted as a pseudo-Goldstone mode of the spontaneous SO(5) symmetry breaking.
The Hubbard Hamiltonian on a two-leg ladder is studied numerically using quantum Monte Carlo and Exact Diagonalization techniques. A rung interaction, VVV, is turned on such that the resulting model has an exact SO(5) symmetry when V=−UV=-UV=−U. The evolution of the low energy excitation spectrum is presented from the pure Hubbard ladder to the SO(5) ladder. It is shown that the low energy excitations in the pure Hubbard ladder have an approximate SO(5) symmetry.
The Hubbard Hamiltonian on a two-leg ladder is studied numerically using quantum Monte Carlo and exact diagonalization techniques. A rung interaction, V,V,V, is turned on such that the resulting model has an exact SO(5) symmetry when $V=\ensuremath{-}U.$ The evolution of the low-energy excitation spectrum is presented from the pure Hubbard ladder to the SO(5) ladder. It is shown that the low-energy excitations in the pure Hubbard ladder have an approximate SO(5) symmetry.