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Physical Review Letters

Phase diagram of the two-dimensional negative- U Hubbard model

Richard T. Scalettar, Eugene Loh, James E. Gubernatis, Adriana Moreo, Steven R. White, Douglas J. Scalapino, Robert L. Sugar, Elbio R. Dagotto

Published Mar 20, 1989• 290 citations
Publisher

Abstract

Theoretical arguments and numerical calculations are used to discuss the phase diagram of the two-dimensional negative-U Hubbard model. Our results are consistent with (1) a vanishing transition temperature at half-filling but with a ground state having both superconducting and charge-density-wave long-range order, and (2) a Kosterlitz-Thouless transition at a finite temperature into a superconducting state with power-law decay of the pairing correlations away from half-filling.

DOI
10.1103/physrevlett.62.1407

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Theoretical arguments and numerical calculations are used to discuss the phase diagram of the two-dimensional negative-U Hubbard model. Our results are consistent with (1) a vanishing transition temperature at half-filling but with a ground state having both superconducting and charge-density-wave long-range order, and (2) a Kosterlitz-Thouless transition at a finite temperature into a superconducting state with power-law decay of the pairing correlations away from half-filling.
Quantum Monte Carlo simulations are used to explore the phase diagram of the negative-U Hubbard model. As the system is doped away from half filling, 〈n〉=1, the s-wave pair-field correlations are enhanced and the charge-density-wave correlations suppressed. There is no indication of phase separation. Away from half filling, the pair-field correlations are consistent with Kosterlitz-Thouless scaling and a finite Tc{\mathit{T}}_{\mathit{c}}Tc​ which peaks near 〈n〉=1 but vanishes at exactly half filling.
We report on a numerical study of the two-dimensional Hubbard model and describe two new algorithms for the simulation of many-electron systems. These algorithms allow one to carry out simulations within the grand canonical ensemble at significantly lower temperatures than had previously been obtained and to calculate ground-state properties with fixed numbers of electrons. We present results for the two-dimensional Hubbard model with half- and quarter-filled bands. Our results support the existence of long-range antiferromagnetic order in the ground state at half-filling and its absence at quarter-filling. Results for the magnetic susceptibility and the momentum occupation along with an upper bound to the spin-wave spectrum are given. We find evidence for an attractive effective d-wave pairing interaction near half-filling but have not found evidence for a phase transition to a superconducting state.
Extensive numerical studies have demonstrated that the two-dimensional single-band Hubbard model contains much of the key physics in cuprate high-temperature superconductors. However, there is no definitive proof that the Hubbard model truly possesses a superconducting ground state or, if it does, of how it depends on model parameters. To answer these longstanding questions, we study an extension of the Hubbard model including an infinite-range ddd-wave pair field term, which precipitates a superconducting state in the ddd-wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the ddd-wave pair/spin/charge-density matrices. The calculations show a ddd-wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion UUU and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating ddd-wave superconductivity.
In order to investigate possible nonphonon mechanisms for superconductivity in the copper oxide compounds, the two-dimensional Hubbard model is studied in the weak-coupling limit. A scaling theory is developed which allows a consistent treatment of the coupled ln 2 ( T )-singularities arising from density wave and Cooper pairing fluctuations. For a half-filled band a spin-density-wave (antiferromagnetic) phase transition occurs, whereas for small deviations from half-filling spin fluctuations lead to a d -type superconducting phase. The method generalizes straightforwardly to finite-range interactions. The possible relevance to oxide superconductors and experimental tests are briefly discussed.
The two-dimensional Hubbard model near half-filling is studied using a configuration-interaction method. Superconducting order-parameter and spin-spin correlation functions are calculated as functions of distance to probe for superconductivity and antiferromagnetism. Consistent with our previous results, these more extensive calculations point toward d-wave superconductivity in the ground state of the two-dimensional repulsive Hubbard model. The d-wave correlations tend to increase with interaction strength, size of the state space used in the calculation, and doping away from half-filling. The superconducting correlations on the Fermi surface are interpreted as d-wave pairing between spin density-wave quasiparticles.
We study the ground state of the two-dimensional Anderson-Hubbard model using a quantum real space renormalization group method. We obtain the phase diagram near half filling. The system is always insulating with disorder. At half filling, the system undergoes a transition from a gapless (Anderson) insulator to an incompressible (Mott-Hubbard) insulator as the interaction UUU reaches a critical value UcU_cUc​. Away from half filling, the insulating phase is always gapless and is found to be controlled by the Anderson fixed point at half filling. This result is similar to that obtained in the corresponding one dimensional system and suggests strongly the importance of the electron-electron correlation in this gapless insulating phase.