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Antiferromagnetism, charge density wave, and d-wave superconductivity in the $t$-$J$-$U$-$V$ model of correlated electrons: Role of direct Coulomb interactions

Marcin Abram, M. Zegrodnik, Józef Spałek

Published Jul 19, 2016• 0 citations
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Abstract

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 using the statistically consistent Gutzwiller approximation (SGA). We concentrate on the role of the intersite Coulomb interaction term $V$ in stabilizing the CDW phase and show it appears only above some critical values of $V$ and in a limited hole doping range $\delta$. The effect of the term $V$ 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.

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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.
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.
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.
We study the coexistence of pair- (PDW) and charge-density-wave (CDW) states within the single-band $t\text{\ensuremath{-}}J\text{\ensuremath{-}}U$ and Hubbard models of ddd-wave superconductivity and discuss our results in the context of the experimental observations for the copper-based compounds. In order to take the correlation effects into account with proper precision, we use the approach based on the diagrammatic expansion of the Gutzwiller wave function (DE-GWF), which goes beyond the renormalized mean-field theory in a systematic manner. According to our analysis of the $t\text{\ensuremath{-}}J\text{\ensuremath{-}}U$ model, the transition between the pure ddd-wave superconducting phase and the coexistent CDW+PDW phase takes place at $\ensuremath{\delta}\ensuremath{\approx}0.18$ (close to the optimal doping), with the modulated phase located in the underdoped regime. The situation is different for the case of the Hubbard model, where a narrow stability regime of a precursor nematic phase sets in preceding the formation of the modulated CDW+PDW state, with decreasing hole doping. The results conclude our comprehensive discussion of the standard phase diagram for high-TC{T}_{C}TC​ superconducting compounds within the DE-GWF variational approach in the single narrow-band case [see J. Spa\l{}ek et al., Phys. Rev. B 95, 024506 (2017); M. Zegrodnik and Spa\l{}ek, Phys. Rev. B 95, 024507 (2017); Phys. Rev. B 96, 054511 (2017); New J. Phys. 20, 063015 (2018)].
We study the effect of the correlated hopping term and the intersite Coulomb interaction term on principal features of the ddd-wave superconducting (SC) state, in both the electron- and hole-doped regimes within the $t\text{\ensuremath{-}}J\text{\ensuremath{-}}U$ model. In our analysis, we use the approach based on the diagrammatic expansion of the Gutzwiller wave function (DE-GWF), which allows us to go beyond the renormalized mean-field theory (RMFT). We show that the correlated hopping term enhances the pairing at the electron-doped side of the phase diagram. Moreover, the so-called non-BCS regime (which manifests itself by the negative kinetic energy gain at the transition to the SC phase) is narrowed down with the increasing magnitude of the correlated hopping $\ensuremath{\sim}K$. Also, the doping dependencies of the nodal Fermi velocity and Fermi momentum, as well as the average number of double occupancies, are analyzed with reference to the experimental data for selected values of the parameter KKK. For the sake of completeness, the influence of the intersite Coulomb repulsion on the obtained results is provided. Additionally, selected results concerning the Hubbard-model case are also presented. A complete model with all two-site interactions is briefly discussed in Appendix for reference.
We analyze the effect of intersite interactions on the stability of the coexisting superconducting-nematic phase (SC+N) within the extended Hubbard and t – J – U models on the square lattice. In order to take into account the correlation effects to a proper precision, we use the approach based on the diagrammatic expansion of the Gutzwiller wave function (DE-GWF), which goes beyond the renormalized mean-field theory (RMFT) in a systematic manner. As a starting point of our analysis we discuss the SC+N phase stability as a function of the intrasite Coulomb repulsion and hole doping for the case of the Hubbard model. Next, we show that the exchange interaction term enhances superconductivity while suppresses the nematicity, whereas the intersite Coulomb repulsion acts in the opposite manner. The competing character of the SC and N phases interplay is clearly visible throughout the analysis. A universal conclusion is that the nematic phase does not survive within the t – J – U model for the value of J integral typical for the high- T C cuprates ( J ≈ 0.1 eV). This result is helpful in providing the understanding of the fundamental role of the nodal direction. For the sake of completeness, the effect of the correlated hopping term is also analyzed. Thus the present discussion contains all relevant two-site interactions which appear in the parametrized single-band model of correlated fermions. At the end, the influence of the higher-order terms of the DE on the rotational symmetry breaking is also shown by comparing the DE-GWF results with those of the RMFT.
Motivated by significant discrepancies between experimental observations of electron-doped cuprates and numerical results of the Hubbard and $t\text{\ensuremath{-}}J$ models, we investigate the role of intersite interactions VVV by studying the $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ model on square lattices. Based on large-scale density matrix renormalization group simulations, we identify the ground-state phase diagram across varying intersite interactions VVV and doping concentration $\ensuremath{\delta}$. We find that the phase diagram with finite intersite interactions $2\ensuremath{\lesssim}V/J\ensuremath{\lesssim}3$ offers a more accurate description of electron-doped cuprates than the conventional Hubbard and $t\text{\ensuremath{-}}J$ models. Moreover, we reveal the role of intersite interactions VVV at varying doping levels: at light doping, intersite interactions favor N\'eel antiferromagnetic order, and suppress both superconductivity and charge density wave; around optimal doping, these interactions support a pseudogap-like phase while suppressing superconductivity, and we further perform the slave-boson mean-field analysis to understand the numerical results microscopically; at higher doping, the effects of intersite interactions become insignificant, with our numerical predictions suggesting the emergence of an incommensurate spin density wave phase. Our specific focus around optimal doping with various intersite interactions identifies successive phases including phase separation, uniform ddd-wave SC, and a pseudogap-like phase, and reveals a relative insensitivity of the charge density wave to superconductivity. Our study suggests the $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ model as the minimal model to capture the essential physics of electron-doped cuprates.