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Physical review. B, Condensed matter

Effect of the intersite Coulomb interaction in the extended Hubbard model

Mei-Rong Li, Yong-Jun Wang, Chang-De Gong

Published Aug 1, 1992• 1 citation
Publisher

Abstract

Starting from the large-U Hubbard model combining the intersite Coulomb interaction V, we investigate the competition, on a square lattice, between the d-wave resonating-valence-bond (RVB) state and the staggered-flux-phases (SFP) state on the basis of F. C. Zhang's earlier work [Phys. Rev. Lett. 64, 974 (1990)]. We find that a very small V (0.1U) may help stabilize the SFP state by effectively suppressing the d-wave pairing amplitude and enhance the particle-hole correlation. We show in the phase diagram of V-\ensuremath{\delta} (the doping concentration) the critical curve of ${\mathit{V}}_{\mathit{c}}$(\ensuremath{\delta}) below which the RVB state is energetically favored while beyond this point a mixed state of SFP and RVB occurs, which has the lowest energy. We also find that the pure SFP state is not the most stable state in the whole V region.

DOI
10.1103/physrevb.46.3668

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Starting from the large-U Hubbard model combining the intersite Coulomb interaction V, we investigate the competition, on a square lattice, between the d-wave resonating-valence-bond (RVB) state and the staggered-flux-phases (SFP) state on the basis of F. C. Zhang's earlier work [Phys. Rev. Lett. 64, 974 (1990)]. We find that a very small V (0.1U) may help stabilize the SFP state by effectively suppressing the d-wave pairing amplitude and enhance the particle-hole correlation. We show in the phase diagram of V-\ensuremath{\delta} (the doping concentration) the critical curve of Vc{\mathit{V}}_{\mathit{c}}Vc​(\ensuremath{\delta}) below which the RVB state is energetically favored while beyond this point a mixed state of SFP and RVB occurs, which has the lowest energy. We also find that the pure SFP state is not the most stable state in the whole V region.
The authors have numerically evaluated the energy of the staggered flux phase state (SFP), the d-wave resonating valence bond state, and the mixed SFP and d-wave state (SFP+d) of the extended t-J model at various hole densities ( delta ). The flux per plaquette and the size of the d-wave gap are taken as variational parameters. The calculation includes the kinetic energy, spin energy, the effective hole-hole interaction and the three-site pair-hopping term. They found that the SFP is stable against the projected Fermi liquid up to delta approximately=20%. For moderately large J, J/t>or=0.4, the SFP+d state may be favoured against the pure d-wave state but only at very low hole density ( delta approximately 20%). These calculations show that the pure SFP state is never the ground state at hole densities of experimental interest.
Motivated by the observation of inhomogeneous patterns in some high-Tc{T}_{c}Tc​ cuprate compounds, several variational Gutzwiller-projected wave functions with built-in charge and bond-order parameters are proposed for the extended $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ model on the square lattice at low doping. First, following a recent Gutzwiller-projected mean-field approach by one of us [D. Poilblanc, Phys. Rev. B 72, 060508(R) (2005)], we investigate, as a function of doping and Coulomb repulsion the relative stability of a wide variety of modulated structures with square unit cells of size $2\ifmmode\times\else\texttimes\fi{}2$, $\sqrt{8}\ifmmode\times\else\texttimes\fi{}\sqrt{8}$, $4\ifmmode\times\else\texttimes\fi{}4$, and $\sqrt{32}\ifmmode\times\else\texttimes\fi{}\sqrt{32}$. It is found that the $4\ifmmode\times\else\texttimes\fi{}4$ bond-order wave function with staggered-flux pattern (and small charge and spin current density wave) is a remarkable competitive candidate for hole doping around 1∕81∕81∕8 in agreement with scanning tunneling microscopy observations in the underdoped regime of some cuprates. This wave function is then optimized accurately and its properties studied extensively using a variational Monte Carlo scheme. Moreover, we find that under increasing the Coulomb repulsion, the ddd-wave superconducting RVB wave function is rapidly destabilized with respect to the $4\ifmmode\times\else\texttimes\fi{}4$ bond-order wave function. The stability of the bond-modulated wave function is connected to a gain of Coulomb and exchange energies. We suggest that such ordering patterns could be dynamical or could spontaneously appear in the vicinity of an impurity or a vortex in the mixed phase of the cuprates. Finally, we consider also a commensurate flux phase, but this wave function turns out not to be competitive because of its rather poor kinetic energy. However, we find it has very competitive exchange and Coulomb energies.
We have calculated the energy of the resonating-valence-bond (RVB) state made of nearest-neighbor singlet pairs in the two-dimensional Hubbard model at half filling and the energy of holes in the RVB state near half filling. The energy of the RVB state was obtained by extrapolating the result in a 6\ifmmode\times\else\texttimes\fi{}6 square lattice with the periodic boundary condition to the infinite lattice. The energy of holes was obtained by examining the first five moments of the density of states of holes in the RVB state. We find that the holes are more stabilized in the RVB state than in the N\'eel state for J/t\ensuremath{\gtrsim}0.1, where J and t are the exchange interaction and the nearest-neighbor hopping integral, respectively. We discuss the possibility of the RVB state in the high-Tc{T}_{c}Tc​ Cu oxides.
The competition and possible coexistence of the flux phase and pairing states are considered within the slave-boson approach. In addition to the gauge field, coupling holons, and spinons, a statistical gauge field is introduced to account for the hard-core nature of holons. Using the equivalence of \ensuremath{\pi}-uniform and -staggered fluxes, we show the possibility of reducing the staggered flux and guaranteeing the commensurability condition of the uniform flux for holons at the same time. For certain dopings the frustration of the spinon gauge field to holon motion can be compensated by the holon's own statistical gauge field. Contrary to earlier conclusions of other authors, we find that the flux phase can coexist with the resonating-valence-bond (RVB) states below a critical doping concentration ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$, beyond which the d-wave RVB state takes over.
We describe a new method to numerically evaluate the properties of correlated superconducting wave functions. We have applied it to the resonating-valence-bond (RVB) wave function for the Hubbard model on the square lattice. For the half-filled case we find that the ddd-wave RVB state and the antiferromagnetic ordered state have the same energy within numerical accuracy. At 10% doping we find ddd-wave superconductivity, consistent with previous studies. We show that the superconducting order parameter is proportional to the number of holes, for small hole concentrations.