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arXiv:cond-mat/0002352 · Physical Review Letters

d -Wave Superconductivity in the Hubbard Model

Thomas Maier, Mark Jarrell, Thomas Pruschke, J. Keller

Published Aug 14, 2000• 156 citations
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Abstract

The superconducting instabilities of the doped repulsive 2D Hubbard model are studied in the intermediate to strong coupling regime with the help of the dynamical cluster approximation. To solve the effective cluster problem we employ an extended noncrossing approximation, which allows for a transition to the broken symmetry state. At sufficiently low temperatures we find stable d-wave solutions with off-diagonal long-range order. The maximal T(c) approximately 150 K occurs for a doping delta approximately 20% and the doping dependence of the transition temperatures agrees well with the generic high- T(c) phase diagram.

DOI
10.1103/physrevlett.85.1524

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The superconducting instabilities of the doped repulsive 2D Hubbard model are studied in the intermediate to strong coupling regime with the help of the dynamical cluster approximation. To solve the effective cluster problem we employ an extended noncrossing approximation, which allows for a transition to the broken symmetry state. At sufficiently low temperatures we find stable d-wave solutions with off-diagonal long-range order. The maximal T(c) approximately 150 K occurs for a doping delta approximately 20% and the doping dependence of the transition temperatures agrees well with the generic high- T(c) phase diagram.
Using an extended dynamical cluster approximation we study superconductivity in the two-dimensional t-J model. In analogy to the extended dynamical mean field theory, non-local spin fluctuations are treated self-consistently with an additional Bosonic bath. To solve the effective cluster problem, we employ an extended non-crossing approximation which allows for transitions to the symmetry-broken state. At sufficiently low temperatures we find a stable superconducting solution with d-wave order parameter. Upon pairing, the exchange energy is lowered, consistent with an exchange-based pairing mechanism.
The cluster size dependence of superconductivity in the conventional two-dimensional Hubbard model, commonly believed to describe high-temperature superconductors, is systematically studied using the dynamical cluster approximation and quantum Monte Carlo simulations as a cluster solver. Because of the nonlocality of the d-wave superconducting order parameter, the results on small clusters show large size and geometry effects. In large enough clusters, the results are independent of the cluster size and display a finite temperature instability to d-wave superconductivity.
We compute the two-particle quantities relevant for superconducting correlations in the two-dimensional Hubbard model within the dynamical cluster approximation. In the normal state we identify the parameter regime in density, interaction, and second-nearest-neighbor hopping strength that maximizes the d_{x^{2}-y^{2}} superconducting transition temperature. We find in all cases that the optimal transition temperature occurs at intermediate coupling strength, and is suppressed at strong and weak interaction strengths. Similarly, superconducting fluctuations are strongest at intermediate doping and suppressed towards large doping and half filling. We find a change in sign of the vertex contributions to d_{xy} superconductivity from repulsive near half filling to attractive at large doping. p-wave superconductivity is not found at the parameters we study, and s-wave contributions are always repulsive. For negative second-nearest-neighbor hopping the optimal transition temperature shifts towards the electron-doped side in opposition to the van Hove singularity, which moves towards hole doping. We surmise that an increase of the local interaction of the electron-doped compounds would increase T_{c}.
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.
We study the competition of antiferromagnetism and ddd-wave superconductivity at zero temperature in the two-dimensional Hubbard model using cellular dynamical mean-field theory for a $2\ifmmode\times\else\texttimes\fi{}2$ plaquette, and solve the associated cluster impurity model at zero temperature by means of exact diagonalization. The interplay between the two phases depends strongly on the strength of the correlation. At strong coupling $(U\ensuremath{\gtrsim}8t)$ the two phases do not mix, and a first-order transition takes place as a function of doping between two pure phases. At weak coupling $(U\ensuremath{\lesssim}8t)$ the two order parameters coexist within the same solution in a range of doping and the system smoothly evolves from the antiferromagnet to the superconductor. When the transition between the superconducting and the antiferromagetic phases is of the first-order, it is accompanied by a phase separation.
We present an approach to investigate the interplay of antiferromagnetism and d-wave superconductivity in the two-dimensional Hubbard model within a numerically exact cluster dynamical mean-field approximation. Self-consistent solutions with two nonzero order parameters exist in a wide range of doping level and temperatures. A linearized equation for the energy spectrum near the Fermi level has been solved. The resulting d-wave gap has the correct magnitude and k dependence, but some distortion compared to the pure ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconducting order parameter due to the presence of underlying antiferromagnetic ordering.