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Scientific Reports

Superconducting and charge-ordered states in the anisotropic t–J–U model

Yifan Feng, Jie Lou, Yan Chen

Published Jan 16, 2024• 1 citation
Publisher

Abstract

Motivated by the effect of symmetry breaking in cuprates superconductors YBa[Formula: see text]Cu[Formula: see text]O[Formula: see text], we employ the renormalized mean-field theory to study the presence of uniform superconducting and charge-ordered states in two anisotropic t-J-U models, either with hopping strength anisotropy or antiferromagnetic interaction anisotropy. In the case of uniform superconducting state, compared with the isotropic t-J-U model with only [Formula: see text]-wave superconducting state, there is an additional s-wave superconducting state in the model with hopping strength anisotropy. Meanwhile, the hopping anisotropy may enhance the critical Coulomb interaction [Formula: see text] at the Mott insulator to the Gossamer superconductor transition point, and strong hopping anisotropy may weaken the superconducting state. In the case of a charge-ordered state, hopping anisotropy may suppress the amplitude of the charge density waves and pair density waves, which originate from local Coulomb interactions. These results indicate that the effects of hopping anisotropy and local Coulomb interactions are competitive. Moreover, the antiferromagnetic interaction anisotropy only weakly suppresses the superconducting gap and density wave amplitude. Our results show that the t-J-U model with hopping anisotropy is qualitatively consistent with experimental superconducting pair symmetry and charge density waves in the YBa[Formula: see text]Cu[Formula: see text]O[Formula: see text] system.

DOI
10.1038/s41598-024-51829-7

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Motivated by the effect of symmetry breaking in cuprates superconductors YBa[Formula: see text]Cu[Formula: see text]O[Formula: see text], we employ the renormalized mean-field theory to study the presence of uniform superconducting and charge-ordered states in two anisotropic t-J-U models, either with hopping strength anisotropy or antiferromagnetic interaction anisotropy. In the case of uniform superconducting state, compared with the isotropic t-J-U model with only [Formula: see text]-wave superconducting state, there is an additional s-wave superconducting state in the model with hopping strength anisotropy. Meanwhile, the hopping anisotropy may enhance the critical Coulomb interaction [Formula: see text] at the Mott insulator to the Gossamer superconductor transition point, and strong hopping anisotropy may weaken the superconducting state. In the case of a charge-ordered state, hopping anisotropy may suppress the amplitude of the charge density waves and pair density waves, which originate from local Coulomb interactions. These results indicate that the effects of hopping anisotropy and local Coulomb interactions are competitive. Moreover, the antiferromagnetic interaction anisotropy only weakly suppresses the superconducting gap and density wave amplitude. Our results show that the t-J-U model with hopping anisotropy is qualitatively consistent with experimental superconducting pair symmetry and charge density waves in the YBa[Formula: see text]Cu[Formula: see text]O[Formula: see text] system.
We study two charge-ordered states in a $t\ensuremath{-}J\ensuremath{-}U$ model with the method of renormalized mean-field theory. Compared with the $t\ensuremath{-}J$ model, which excludes the existence of one site occupied by two electrons, finite Coulomb interaction UUU will allow the appearance of doublons and produce more charge fluctuations. The charge density wave states accompanied by Cooper pair density modulations have similar features with the states found in the $t\ensuremath{-}J$ model. The difference is that the modulation of doublons exists, and it has the same wavelength and phase as holes. Charge-ordered states disappear at the $U/t\ensuremath{\le}9$ region when the system does not correspond to the strongly correlated resonating valence-bond state. This feature indicates that local Coulomb interactions are the origin of charge-ordered states observed in cuprates. In addition, we calculate the strength of superconducting coherence and the gap depth of the nodal pair-density-wave state based on the local density of states. These two quantities also vary with lattice sites periodically. The oscillations have same wave vector and phase as the modulation of holes. These results are consistent with recent experiments.
We study the charge-density-wave order and its competition with superconductivity in the one-band Hubbard model for high-Tc{T}_{c}Tc​ superconducting cuprates. We use cluster dynamical mean field theory at T=0T=0T=0. The one-band Hubbard model contains only copper atoms, and a physical charge-density wave with charge excesses located on the oxygen atoms manifests itself as a bond-density wave (BDW) within this context. It arises purely out of local correlation effects and also leads to an ${s}^{\ensuremath{'}}$-wave pair-density wave in the presence of ddd-wave superconductivity. The ddd-wave BDW is suppressed on increasing UUU and is favored on increasing the magnitude of the second-neighbor hopping ${t}^{\ensuremath{'}}$. Further, the ddd-wave BDW order is weakened when in competition with superconductivity, as seen in experiments. Additionally, it behaves quite differently on varying UUU in the superconducting state compared to the normal state.
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.
Unravelling competing orders emergent in doped Mott insulators and their interplay with unconventional superconductivity is one of the major challenges in condensed matter physics. To explore the possible superconducting state in a doped Mott insulator, we study the square-lattice t-J model with both the nearest-neighbor and next-nearest-neighbor electron hoppings and spin interactions. By using the state-of-the-art density matrix renormalization group calculation with imposing charge U(1) and spin SU(2) symmetries on the six-leg cylinders, we establish a quantum phase diagram including three phases: a stripe charge density wave phase, a superconducting phase without static charge order, and a superconducting phase coexistent with a weak charge stripe order. Crucially, we demonstrate that the superconducting phase has a power-law pairing correlation that decays much slower than the charge density and spin correlations, which is a quasi-1D descendant of the uniform d-wave superconductor in two dimensions. These findings reveal that enhanced charge and spin fluctuations with optimal doping is able to produce robust d-wave superconductivity in doped Mott insulators, providing a foundation for connecting theories of superconductivity to models of strongly correlated systems.
Antiferromagnetism, charge density wave, and d-wave superconductivity in the ttt-JJJ-UUU-VVV model of correlated electrons: Role of direct Coulomb interactions
We study the stability of antiferromagnetic (AF), charge density wave (CDW), and superconducting (SC) states within the ttt-JJJ-UUU-VVV model of strongly correlated electrons using the statistically consistent Gutzwiller approximation (SGA). We concentrate on the role of the intersite Coulomb interaction term VVV in stabilizing the CDW phase and show it appears only above some critical values of VVV and in a limited hole doping range δ\deltaδ. The effect of the term VVV on SC and AF phases is that a strong interaction suppresses SC altogether, whereas the AF order is not significantly influenced by its presence. Calculations for the case of pure SC phase have been also carried out within an extended approach, the extended approach, the diagrammatic expansion for the Gutzwiller wave function (DE-GWF) in order to analyze influence of the higher-order correlations onto this phase beyond the renormalized mean field approach. In the Appendices we discuss, the specific ambiguity of the choice of the Gutzwiller renormalization factors within the SGA when either AF or CDW orders are considered.
The one-band t–J model captures strong correlations in cuprate high-temperature superconductors. It accounts for the various intertwined spin and charge orders, and the superconductivity in the phase diagrams. To see the correlation effect on the intertwined orders, we implement the density matrix renormalization group method to simulate the t–J model in a small J case with t/J = 10, which is in a deeper Mott region than that with t/J ≃ 3 in cuprate superconducting compounds. We examine the results on a six-leg lattice with both the nearest and next-nearest-neighbor hoppings and antiferromagnetic coupling, and find the absence of superconductivity and enhanced intertwined spin and charge orders in the phase diagram. Besides the stripe phases, we find a new SDW + CDW phase in which the spin modulation is a (π, π) antiferromagnetism, while the wavelength of the charge modulation is shorter than that of the stripe phases. Our results suggest the enhanced intertwined orders and suppressed superconductivity in the deep Mott region.