We analyze the competition between antiferromagnetism and superconductivity in the two-dimensional Hubbard model by combining a functional renormalization group flow with a mean-field theory for spontaneous symmetry breaking. Effective interactions are computed by integrating out states above a scale ${\ensuremath{\Lambda}}_{\mathrm{MF}}$ in one-loop approximation, which captures in particular the generation of an attraction in the
d-wave Cooper channel from fluctuations in the particle-hole channel. These effective interactions are then used as an input for a mean-field treatment of the remaining low-energy states, with antiferromagnetism, singlet superconductivity, and triplet $\ensuremath{\pi}$ pairing as the possible order parameters. Antiferromagnetism and superconductivity suppress each other, leaving only a small region in parameter space where both orders can coexist with a sizable order parameter for each. Triplet $\ensuremath{\pi}$ pairing appears generically in the coexistence region, but its feedback on the other order parameters is very small.
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.
We study the competition of antiferromagnetism and
d-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 discuss the coexistence of antiferromagnetism and
d-wave superconductivity within the so-called statistically consistent Gutzwiller approximation (SGA) applied to the
t--
J--
U model. In this approach, the averages calculated in a self-consistent manner coincide with those determined variationally. Such consistency is not guaranteed within the standard renormalized mean-field theory. With the help of SGA, we show that for the typical value
J/∣t∣=31, coexistence of antiferromagnetism (AF) and superconductivity (SC) appears only for
U/∣t∣>10.6 and in a very narrow range of doping $(\ensuremath{\delta}\ensuremath{\lesssim}0.006)$ in the vicinity of the Mott insulating state, in contrast to some previous reports. In the coexistent AF+SC phase, a staggered spin-triplet component of the superconducting gap appears also naturally; its value is very small.
Proximity to a Mott insulating phase is likely to be an important physical ingredient of a theory that aims to describe high-temperature superconductivity in the cuprates. Quantum cluster methods are well suited to describe the Mott phase. Hence, as a step toward a quantitative theory of the competition between antiferromagnetism and
d-wave superconductivity in the cuprates, we use cellular dynamical mean-field theory to compute zero-temperature properties of the two-dimensional square lattice Hubbard model. The
d-wave order parameter is found to scale like the superexchange coupling
J for on-site interaction
U comparable to or larger than the bandwidth. The order parameter also assumes a dome shape as a function of doping, while, by contrast, the gap in the single-particle density of states decreases monotonically with increasing doping. In the presence of a finite second neighbor hopping ${t}^{\ensuremath{'}}$, the zero-temperature phase diagram displays the electron-hole asymmetric competition between antiferromagnetism and superconductivity that is observed experimentally in the cuprates. Adding realistic third neighbor hopping ${t}^{\ensuremath{''}}$ improves the overall agreement with the experimental phase diagram. Since band parameters can vary depending on the specific cuprate considered, the sensitivity of the theoretical phase diagram to band parameters challenges the commonly held assumption that the doping vs
Tc∕Tcmax phase diagram of the cuprates is universal. The calculated angle-resolved photoemission spectrum displays the observed electron-hole asymmetry. The tendency to homogeneous coexistence of the superconducting and antiferromagnetic order parameters is stronger than observed in most experiments but consistent with many theoretical results and with experiments in some layered high-temperature superconductors. Clearly, our calculations reproduce important features of
d-wave superconductivity in the cuprates that would otherwise be considered anomalous from the point of view of the standard Bardeen--Cooper--Schrieffer approach. At strong coupling,
d-wave superconductivity and antiferromagnetism naturally appear as two equally important competing instabilities of the normal phase of the same underlying Hamiltonian.
Interplay between antiferromagnetism (AF) and d-wave superconductivity
(dSC) is investigated in the slave-boson scheme of the two-dimensional $t\ensuremath{-}J$ model on the square lattice. So far, it seems that their coexistence is believed to be a general feature. It is, however, reported in this paper that the coexistence is suppressed significantly by ${t}^{\ensuremath{''}},$ the third neighbor hopping. This effect will lead to noticeable material dependence of the possible bulk coexistence of AF and
dSC.