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arXiv:cond-mat/0604400 · Modern Physics Letters B

INHOMOGENEOUS d-WAVE SUPERCONDUCTIVITY AND ANTIFERROMAGNETISM IN A TWO-DIMENSIONAL EXTENDED HUBBARD MODEL WITH NEAREST-NEIGHBOR ATTRACTIVE INTERACTION

W. P. SU

Published Apr 20, 2007• 1 citation
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Abstract

To understand the interplay of d-wave superconductivity and antiferromagnetism, we consider a two-dimensional extended Hubbard model with nearest neighbor attractive interaction. The Hamiltonian is solved in the mean field approximation on a finite lattice. In the impurity-free case, the minimum energy solutions show phase separation as predicted previously based on free energy argument. The phase separation tendency implies that the system can be easily rendered inhomogeneous by a small external perturbation. Explicit solutions of a model including weak impurity potentials are indeed inhomogeneous in the spin-density-wave and d-wave pairing order parameters. Relevance of the results to the inhomogeneous cuprate superconductors is discussed.

DOI
10.1142/s0217984907013092

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To understand the interplay of d-wave superconductivity and antiferromagnetism, we consider a two-dimensional extended Hubbard model with nearest neighbor attractive interaction. The Hamiltonian is solved in the mean field approximation on a finite lattice. In the impurity-free case, the minimum energy solutions show phase separation as predicted previously based on free energy argument. The phase separation tendency implies that the system can be easily rendered inhomogeneous by a small external perturbation. Explicit solutions of a model including weak impurity potentials are indeed inhomogeneous in the spin-density-wave and d-wave pairing order parameters. Relevance of the results to the inhomogeneous cuprate superconductors is discussed.
Inhomogeneities on various length scales suggesting phase separation seem to be a generic feature of hole-doped high-temperature superconducting cuprates. Also firmly established is the d-wave symmetry of the superconducting state. The interplay of superconductivity and phase separation has been noted in previous mean-field studies of correlated electron models. To further explore this issue in a more rigorous approach we have conducted quantum Monte Carlo calculations of a two-dimensional Hubbard model with nearest-neighbor attractive interaction. For a vanishing on-site repulsion, the result agrees with that of Micnas, Ranninger, and Robaszkiewicz [Phys. Rev. B 39, 11 653 (1989)], and indicates that for a given strength of the attractive interaction, homogeneous d-wave superconducting phase can exist only above a critical doping concentration, below which phase separation occurs. A finite on-site repulsion modifies the phase separation behavior. A stronger d-wave pairing correlation is seen in the inhomogeneous phase compared to the homogeneous phase.
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.
Our previous mean-field study concerning superconducting pairings in the extended Hubbard model with on-site repulsive and intersite attractive interaction and arbitrary electron density is extended to include the effects of nearest-neighbor exchange (JJJ) and correlated hopping (KKK) interaction. Detailed numerical analysis of sss-, ppp-, and ddd-wave pairing solutions is performed for the two-dimensional square lattice with nearest- (ttt) and next-nearest- (t2{t}_{2}t2​) neighbor hopping, and the resulting mutual stability phase diagrams are given as a function of band filling and the interaction parameters. Antiferromagnetic exchange can enhance both the ddd-wave and the extended sss-wave pairing. If it prevails over the repulsive intersite density-density interaction, such a term can be a leading mechanism for superconductivity. However, its effects strongly depend on the band filling and the t2t\frac{{t}_{2}}{t}tt2​​ ratio, giving in particular for t2=0{t}_{2}=0t2​=0 sss-wave (ddd-wave) pairing as most favorable in the low (high) density limit, but increasing t2{t}_{2}t2​ can even reverse this tendency. The correlated hopping term which breaks the electron-hole symmetry affects sss-wave pairing mostly. A condensation transition (phase separation), possible in the presence of the attractive density-density interaction, is also analyzed within the random-phase approximation, and stability conditions of such an electron droplet phase with respect to other types of ordering are determined as a function of the band filling for the two-dimensional square lattice.
Cooperation and competition between the antiferromagnetic, ddd-wave superconducting, and Mott-insulating states are explored for the two-dimensional Hubbard model including nearest and next-nearest-neighbor hoppings at zero temperature. Using the variational cluster approach with clusters of different shapes and sizes up to 10 sites, it is found that the doping-driven transition from a phase with microscopic coexistence of antiferromagnetism and superconductivity to a purely superconducting phase is discontinuous for strong interaction and accompanied by phase separation. At half-filling the system is in an antiferromagnetic Mott-insulating state with vanishing charge compressibility. Upon decreasing the interaction strength UUU below a certain critical value of roughly ${U}_{c}\ensuremath{\sim}4$ (in units of the nearest-neighbor hopping), however, the filling-dependent magnetic transition changes its character and becomes continuous. Phase separation or, more carefully, the tendency towards the formation of inhomogeneous states disappears. This critical value is in contrast to previous studies, where a much larger value was obtained. Moreover, we find that the system at half-filling undergoes the Mott transition from an insulator to a state with a finite charge compressibility at essentially the same value. The weakly correlated state at half-filling exhibits superconductivity microscopically admixed to the antiferromagnetic order. This scenario suggests a close relation between phase separation and the Mott-insulator physics.
Inhomogeneous sss-wave superconductivity is studied in the two-dimensional, square lattice attractive Hubbard Hamiltonian using the Bogoliubov-de Gennes (BdG) mean field approximation. We find that at weak coupling, and for densities mainly below half-filling, an inhomogeneous interaction in which the on-site interaction Ui{U}_{i}Ui​ takes on two values, Ui=0,2U{U}_{i}=0,\phantom{\rule{0.3em}{0ex}}2UUi​=0,2U results in a larger zero temperature pairing amplitude, and that the superconducting Tc{T}_{c}Tc​ can also be significantly increased, relative to a uniform system with Ui=U{U}_{i}=UUi​=U on all sites. These effects are observed for stripe, checkerboard, and even random patterns of the attractive centers, suggesting that the pattern of inhomogeneity is unimportant. Monte Carlo calculations which reintroduce some of the fluctuations neglected within the BdG approach see the same effect, both for the attractive Hubbard model and a Hamiltonian with ddd-wave pairing symmetry.