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Physical review. B, Condensed matter

Coexistence of dx2−y2 superconductivity and antiferromagnetism in the two-dimensional t−J model and numerical estimation of Gutzwiller factors

Akihiro Himeda, Masao Ogata

Published Oct 1, 1999• 128 citations
Publisher

Abstract

The coexisting phase of the d-wave superconducting and the antiferromagnetic order in the two-dimensional t-$J$ model is reexamined using a variational Monte Carlo method in order to estimate the Gutzwiller factors. A trial wave function is used which is consistent with the mean-field solution. The Gutzwiller factors are determined by comparing the numerically obtained expectation values with the mean-field values. We find the z component of the Gutzwiller factor of exchange interaction is enhanced compared with the $\mathrm{xy}$ components. This effect does not appear in the conventional Gutzwiller approximation and is essential for the stabilization of the antiferromagnetic order in the presence of d-wave superconductivity. We discuss several applications of the present Gutzwiller factors to the problems with antiferromagnetism and d-wave superconductivity.

DOI
10.1103/physrevb.60.r9935

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The coexisting phase of the d-wave superconducting and the antiferromagnetic order in the two-dimensional t-JJJ model is reexamined using a variational Monte Carlo method in order to estimate the Gutzwiller factors. A trial wave function is used which is consistent with the mean-field solution. The Gutzwiller factors are determined by comparing the numerically obtained expectation values with the mean-field values. We find the z component of the Gutzwiller factor of exchange interaction is enhanced compared with the xy\mathrm{xy}xy components. This effect does not appear in the conventional Gutzwiller approximation and is essential for the stabilization of the antiferromagnetic order in the presence of d-wave superconductivity. We discuss several applications of the present Gutzwiller factors to the problems with antiferromagnetism and d-wave superconductivity.
We discuss the coexistence of antiferromagnetism and ddd-wave superconductivity within the so-called statistically consistent Gutzwiller approximation (SGA) applied to the ttt--JJJ--UUU 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∣=13J/|t|=\frac{1}{3}J/∣t∣=31​, coexistence of antiferromagnetism (AF) and superconductivity (SC) appears only for U/∣t∣>10.6U/|t|>10.6U/∣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.
The ground state of two-dimensional t – J model including superconductivity and antiferromagnetism is studied using a new type of analytic estimation of the effect of strong correlation. The method is an extension of the Gutzwiller approximation which gives renormalization of parameters, t and J , due to the Gutzwiller projection operator excluding double occupancies. It is shown that the essential point of the renormalization is its dependence on the expectation values of superconductivity and antiferromagnetic moment. In special the renormalization factor is anisotropic in spin space, which leads to a reliable estimate of antiferromagnetic order parameters in the phase diagram, in contrast to the conventional slave-boson mean-field theory. The obtained phase diagram shows the coexistence of antiferromagnetism and d x 2 - y 2 -wave superconductivity for the doping rate, δ, less than 10%. At half-filling, the renormalization of the J -term represents some aspects of quantum spin nature. For the finite doping case, there are two kinds of superconducting order parameters. The variational order parameter of the d-wave superconductivity increases as the doping rate decreases, which will correspond to the experimentally observed superconducting gap. In contrast, the actual expectation value of superconducting order parameter is proportional to δ in underdoped region and has a maximum near δ=0.11, which probably determines T c . We also speculate that the ground-state antiferromagnetic order, being destroyed at finite temperatures, gives the energy and temperature scale of the pseudogap. Applications to the inhomogeneous systems such as the vortex state and stripe state are discussed.
A new type of analytic estimation of the effect of strong correlation is developed for the two-dimensional t-J model. It is based on the Gutzwiller approximation which gives the renormalization of parameters, t and J, due to the Gutzwiller's projection operator excluding the double occupancies. The finite-range correlations are taken into account compared with the conventional Gutzwiller approximation where only the on-site expectation values are considered. It is shown that the essential point of the renormalization is its nonlinear dependence on the expectation values of Cooper pairs and antiferromagnetic moment. In particular the renormalization factor of J becomes anisotropic in the presence of antiferromagnetic moment, i.e., that for the z-component is enhanced compared with that for the xy-component. The physical origin of this enhancement is identified as the surrounding antiferromagnetic correlations of a bond. The self-consistency equations for the uniform variational wave functions are derived and solved numerically. Our result gives a reasonable estimate of antiferromagnetic order parameters near half filling, in contrast to the conventional slave-boson mean-field theory and the original Gutzwiller approximation. It is also found that, at half filling, the renormalization of J represents some of the quantum fluctuations of the Heisenberg spin system in a different manner from the spin-wave theory. For finite doping, our results have some similarity to SO(5) symmetric theory. Applications to the inhomogeneous systems such as the vortex state, around nonmagnetic impurities, and stripe state are discussed.
The two-dimensional $t\ensuremath{-}J$ model in a staggered field is studied by exact diagonalization of small clusters. For the low-hole-density region and a realistic value of J/t,J/t,J/t, it is found that the presence of a staggered field strengthens the attraction between two holes. With increasing field, the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconducting correlations are enhanced while the extended-s-wave ones hardly change. This implies that coexistence of the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconducting order and the commensurate antiferromagnetic order occurs in a staggered field.
Band renormalization effects (BRE) are comprehensively studied for a mixed state of dx2-y2-wave superconducting (d-SC) and antiferromagnetic (AF) orders, in addition to simple d-SC, AF, and normal (paramagnetic) states, by applying a variational Monte Carlo method to a two-dimensional Hubbard (t-t'-U) model. In a weakly correlated regime (U=tt'L [t't'L, because the existence of Fermi surfaces near (pi,0) is a requisite for the electron scattering of q=(pi,pi). Actually, the coexistent state appears mainly for t'L/t
In the first part of the paper, we study the stability of antiferromagnetic (AF), charge density wave (CDW), and superconducting (SC) states within the t-J-U-V model of strongly correlated electrons by using the statistically consistent Gutzwiller approximation (SGA). We concentrate on the role of the intersite Coulomb interaction term V in stabilizing the CDW phase. In particular, we show that the charge ordering appears only above a critical value of V in a limited hole-doping range δ. The effect of the V term on SC and AF phases is that a strong interaction suppresses SC, whereas the AF order is not significantly influenced by its presence. In the second part, separate calculations for the case of a pure SC phase have been carried out within an extended approach (the diagrammatic expansion for the Gutzwiller wave function, DE-GWF) in order to analyze the influence of the intersite Coulomb repulsion on the SC phase with the higher-order corrections included beyond the SGA method. The upper concentration for the SC disappearance decreases with increasing V, bringing the results closer to experiment. In appendices A and B we discuss the ambiguity connected with the choice of the Gutzwiller renormalization factors within the renormalized mean filed theory when either AF or CDW orders are considered. At the end, we overview briefly the possible extensions of the current models to put descriptions of the SC, AF, and CDW states on equal footing.