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arXiv:cond-mat/9212008 · Physical Review Letters

Indications of dx2 - y2 superconductivity in the two dimensionalt-Jmodel

Elbio R. Dagotto, José A. Riera

Published Feb 1, 1993• 153 citations
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Abstract

Supercomputing correlations in the two dimensional t-J model at zero temperature are evaluated using numerical techniques. At the fermionic density 〈n〉\ensuremath{\sim}1/2, strong signals of ${\mathit{d}}_{\mathit{x}}^{2}$-${\mathit{y}}^{2}$ superconductivity were observed in the ground state. These conclusions are based on a study of static pairing correlations, the Meissner effect, and flux quantization as indicators of superconductivity. A tentative phase diagram of the two dimensional t-j model is presented.

DOI
10.1103/physrevlett.70.682

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Supercomputing correlations in the two dimensional t-J model at zero temperature are evaluated using numerical techniques. At the fermionic density 〈n〉\ensuremath{\sim}1/2, strong signals of dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 superconductivity were observed in the ground state. These conclusions are based on a study of static pairing correlations, the Meissner effect, and flux quantization as indicators of superconductivity. A tentative phase diagram of the two dimensional t-j model is presented.
Based on the known phenomenology of high-Tc cuprates and the available numerical calculations of the t − J model, a two-dimensional effective fermionic model with the nearest neighbor attraction is proposed. Numerical calculations suggest that the model has the dx2−y2 superconductivity (SC) in the ground state at a low fermionic density. We argue that this model captures the important physics of the dx2−y2 superconducting correlations found earlier in the t − J model by the exact diagonalization approach. Within a self-consistent RPA diagrammatic study, the density and the coupling strength dependence of the critical temperature is calculated. We also investigate the influence of the impurities on our results and show that the suppression of the superconductivity is insignificant, when the retardation effects are accounted for as opposed to the Hartree–Fock approximation.
A two-dimensional fermionic model with attractive interactions is discussed. Its form is motivated by the phenomenology of the underdoped high-Tc{T}_{c}Tc​ cuprates, as discussed in previous literature. The exact solution to the two-particle problem leads to a bound state in the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ subspace. Numerical techniques show that the model has strong ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing correlations in the ground state at low fermionic density. Within a self-consistent random phase approximation diagrammatic study, the model has a ddd-wave superconducting (SC) phase. The density dependence of the critical temperature is calculated. We argue that in the context of ddd-wave SC this model fulfills the role that the attractive on-site Hubbard model has played for sss-wave SC. We also show that another candidate, the attractive $t\ensuremath{-}U\ensuremath{-}V$ model, is actually not useful as a toy model for ddd-wave SC for a variety of reasons.
The equal-time pairing correlation function of the two-dimensional t-J model on a square lattice is studied using a high-temperature expansion method. The sum of the pairing correlation, its spatial dependence, and the correlation length are obtained as functions of temperature down to T≃0.2tT \simeq 0.2 tT≃0.2t. By comparison of single-particle contributions in the correlation functions, we find an effective attractive interaction between quasi-particles in dx2−y2d_{x^2-y^2}dx2−y2​-wave pairings. It is shown that d-wave correlation grows rapidly at low temperatures for 0.5 < n < 0.9, with n being the electron density. The temperature for this growth is roughly scaled by J/2. This is in sharp contrast to the Hubbard model in a weak or intermediate coupling region, where there is no numerical evidence of superconductivity.
Two-dimensional t-J model is studied by a variational Monte Carlo method, using Gutzwiller-Jastrow-type wave functions. Various kinds of superconducting pairing symmetries are compared in order to determine the phase diagram of the ground state in the full J/t-n plane. Near the half filling where the high temperature superconductivity is expected, the d_{x^2-y^2} wave pairing state is always the most stable among various symmetries. The three-site term hardly changes the phase diagram in this regime. In the low electron density, the extended s-type wave becomes a quantitatively good state for large J/t, although the energy gain is small. The Gutzwiller wave function is shown to be the exact ground state in the low-electron-density limit for the supersymmetric case (J/t=2).
We study the t-J model in one and two dimensions. It is shown that superconducting correlations at large distances are enhanced at doping 〈n〉=1/2, and near the region of phase separation. Including a nearest-neighbor density-density repulsion term of strength V, and working at large V/t and J/t, our results can be explained using a previously discussed variational state where electrons are paired in singlets without cluster formation, in analogy to the attractive Hubbard model. We argue that the physics of the t-J-V model is basically independent of the dimension and that a smooth connection exists between small and large V/t. Then, we conjecture that the two-dimensional t-J model should also present strong superconducting correlations near the region of phase separation, at 〈n〉\ensuremath{\sim}1/2. Numerical results both in one and two dimensions support our claims.
The pairing susceptibility of the one-dimensional t-J model is examined numerically by the quantum transfer matrix method. The temperature dependence of the pairing susceptibility χ s shows logarithmic behavior at low temperatures as χ s ∼αlog T for T being the temperature. The coefficient α is determined as a function of J / t and the fillingness for J and t being the exchange interaction and the transfer. Beyond that of noninteracting fermions, α grows rapidly with the increase of | J |.