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arXiv:cond-mat/0701007 · Physical Review B

Unconventional superconducting symmetry in a checkerboard antiferromagnet studied via renormalized mean-field theory

Huaixiang Huang, You‐Quan Li, Jing-Yu Gan, Yan Chen, Fu‐Chun Zhang

Published May 23, 2007• 12 citations
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Abstract

We use a renormalized mean-field theory to study the Gutzwiller projected Bardeen-Cooper-Schrieffer states of the extended Hubbard model in the large $U$ limit or the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J\text{\ensuremath{-}}{J}^{\ensuremath{'}}$ model on a two-dimensional checkerboard lattice. At small ${t}^{\ensuremath{'}}∕t$, the frustration due to the diagonal terms of ${t}^{\ensuremath{'}}$ and ${J}^{\ensuremath{'}}$ does not alter the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave pairing symmetry, and the negative (positive) ${t}^{\ensuremath{'}}∕t$ enhances (suppresses) the pairing order parameter. At large ${t}^{\ensuremath{'}}∕t$, the ground state has an $s\text{\ensuremath{-}}s$ wave symmetry. At the intermediate ${t}^{\ensuremath{'}}∕t$, the ground state is $d+id$ or $d+is$ wave with time reversal symmetry broken.

DOI
10.1103/physrevb.75.184523

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We use a renormalized mean-field theory to study the Gutzwiller projected Bardeen-Cooper-Schrieffer states of the extended Hubbard model in the large UUU limit or the t\ensuremath-t\ensuremath′\ensuremath-J\ensuremath-J\ensuremath′t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J\text{\ensuremath{-}}{J}^{\ensuremath{'}}t\ensuremath-t\ensuremath′\ensuremath-J\ensuremath-J\ensuremath′ model on a two-dimensional checkerboard lattice. At small t\ensuremath′∕t{t}^{\ensuremath{'}}∕tt\ensuremath′∕t, the frustration due to the diagonal terms of t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ and J\ensuremath′{J}^{\ensuremath{'}}J\ensuremath′ does not alter the dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​-wave pairing symmetry, and the negative (positive) t\ensuremath′∕t{t}^{\ensuremath{'}}∕tt\ensuremath′∕t enhances (suppresses) the pairing order parameter. At large t\ensuremath′∕t{t}^{\ensuremath{'}}∕tt\ensuremath′∕t, the ground state has an s\ensuremath-ss\text{\ensuremath{-}}ss\ensuremath-s wave symmetry. At the intermediate t\ensuremath′∕t{t}^{\ensuremath{'}}∕tt\ensuremath′∕t, the ground state is d+idd+idd+id or d+isd+isd+is wave with time reversal symmetry broken.
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 ddd-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 show that the extended attractive Hubbard model on a square lattice allows for a variety of superconducting phases, including exotic mixed-symmetry phases with [Formula: see text] and [Formula: see text] symmetries, and a novel [Formula: see text] state. The calculations are performed within the Hartree-Fock Bardeen-Cooper-Schrieffer framework. The ground states of the mean-field Hamiltonian are obtained via a minimization scheme that relaxes the symmetry constraints on the superconducting solutions, hence allowing for a mixing of s-, p- and d-wave order parameters. The results are obtained within the assumption of uniform-density states. Our results show that extended attractive Hubbard model can serve as an effective model for investigating properties of exotic superconductors.
The ddd-wave superconductivity (dSC) and antiferromagnetism are analytically studied in a renormalized mean-field theory for a two-dimensional $t\text{\penalty1000-\hskip0pt}J$ model plus an on-site repulsive Hubbard interaction UUU. The purpose of introducing the UUU term is to partially impose the no-double-occupancy constraint by employing the Gutzwiller approximation. The phase diagrams as functions of doping $\ensuremath{\delta}$ and UUU are studied. Using the standard value of t∕J=3.0t∕J=3.0t∕J=3.0 and in the large-UUU limit, we show that the antiferromagnetic (AF) order emerges and coexists with the dSC in the underdoped region below the doping $\ensuremath{\delta}\ensuremath{\sim}0.1$. The dSC order parameter increases from zero as the doping increases and reaches a maximum near the optimal doping $\ensuremath{\delta}\ensuremath{\sim}0.15$. In the small-UUU limit, only the dSC order survives while the AF order disappears. As UUU increases up to a critical value, the AF order shows up and coexists with the dSC in the underdoped regime. At half-filling, the system is in the dSC state for small UUU and becomes an AF insulator for large UUU. Within the present mean-field approach, we show that the ground-state energy of the coexistent state is always lower than that of the pure dSC state.
Extended t-t'-t"-J-U models in which the second-nearest-neighbor hopping (t') and third-nearest-neighbor hopping (t") are included are studied using renormalized mean field theory. The models are meant to be low-energy effective models for the Hubbard models, and hence the Heisenberg exchange integral J and Hubbard repulsion U are related by J = 4t(2)/U. The trial wavefunctions for the ground states are partially Gutzwiller projected Hartree-Fock states. The Gutzwiller projection is implemented by means of a Gutzwiller approximation, and the site double occupancy d is taken as a variational parameter. It is found that a large |t'/t| narrows the band filling range that sustains antiferromagnetism (AFM) in the ground state, enhances the d-wave singlet superconductivity (dSC) in hole overdoped systems, but suppresses the dSC in electron overdoped systems. For a system that has large |t'/t| and |t"/t'|, the superconductivity (SC) at the onset of AFM in hole doped band filling is strongly suppressed. On the excitation occurring, when an electron doped system simultaneously contains SC and AFM, the system is found to have a nodeless gap at the Fermi level. Finally, the result of this study is related to experiments on the superconducting cuprates.
We investigate the ionic Hubbard model (IHM) at half-filling in the limit of strong correlations and large ionic potential. The low-energy effective Hamiltonian in this limit, obtained by a similarity transformation, is a modified $t\ensuremath{-}J$ model with effective second-neighbor hopping terms. We explore the possibilities of ddd-wave pairing and extended sss-wave pairing superconducting (SC) phases on a two-dimensional square lattice at zero temperature within a Gutzwiller projected renormalized mean-field theory. In the sector of solutions that forbid spin-ordering, the system shows a finite nonzero ddd-wave as well as extended sss-wave pairing amplitude for $\mathrm{\ensuremath{\Delta}}\ensuremath{\sim}U\ensuremath{\gg}t$. The width of the superconducting phase in $U\ensuremath{-}\mathrm{\ensuremath{\Delta}}$ regime shrinks with increase in UUU and $\mathrm{\ensuremath{\Delta}}$, though the extended sss-wave pairing phase is higher in energy than the ddd-wave pairing superconducting phase. But in a spin-resolved renormalized mean-field calculation, which allows for an antiferromagnetic (AF) order along with the ddd-wave or extended sss-wave pairing, the SC phase is no longer viable and the system shows a direct transition from an AF ordered phase to a paramagnetic band insulator. Except for a thin sliver of a half-metallic AF phase close to the AF transition point, most of the AF ordered phase is a Mott insulator. We benchmarked the AF Mott insulator to band insulator transition within the Gutzwiller projected renormalized mean-field theory against the dynamical mean-field theory solved using continuous time quantum Monte Carlo. Our work suggests that the ground-state phase diagram of the IHM at half-filling in the limit of extreme correlations does not have any SC phase. The SC phase seen in the paramagnetic sector is a metastable phase, being higher in energy than the AF Mott insulator phase.
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.