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arXiv:cond-mat/0503249

4aX4a Structures from Renormalized Mean-field Theory of the t-J Model

Didier Poilblanc

Published Mar 10, 2005• 0 citations
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Abstract

Using the t-J model (which can also include Coulomb repulsion) and the ``plain vanilla'' renormalized mean-field theory of Zhang et al. (1988), stability of inhomogeneous 4a X 4a superstructures as those observed in cuprates superconductors around hole doping 1/8 is investigated. We find a non-uniform 4a X 4a bond order wave involving simultaneously small (non-uniform) plaquette currents of a few ~ 10^{-2} t as well as a small charge density modulation similar to pair density wave order. On the other hand, no supersolid phase involving a decoupling in the superconducting particle-particle channel is found.

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Using the t-J model (which can also include Coulomb repulsion) and the ``plain vanilla'' renormalized mean-field theory of Zhang et al. (1988), stability of inhomogeneous 4a X 4a superstructures as those observed in cuprates superconductors around hole doping 1/8 is investigated. We find a non-uniform 4a X 4a bond order wave involving simultaneously small (non-uniform) plaquette currents of a few ~ 10^{-2} t as well as a small charge density modulation similar to pair density wave order. On the other hand, no supersolid phase involving a decoupling in the superconducting particle-particle channel is found.
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 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.
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.
Motivated by the recently observed pattern of unidirectional domains in high-Tc{T}_{c}Tc​ superconductors [Y. Kohsaka et al., Science 315, 1380 (2007)], we investigate the emergence of spontaneous modulations in the ddd-wave superconducting resonating valence bond phase using the ttt-JJJ model at x=1∕8x=1∕8x=1∕8 doping. Half-filled charge domains separated by four lattice spacings are found to form along one of the crystal axes, leading to modulated superconductivity with out-of-phase ddd-wave order parameters in neighboring domains. Both renormalized mean-field theory and variational Monte Carlo calculations reveal that the energies of modulated and uniform phases are very close to each other.
We carry out the variational Monte Carlo calculation to examine spatially inhomogeneous states in hole- and electron-doped cuprates. By using Gutzwiller approximation, we consider the excitations, arising from charge density, spin density and pair field, of the mean-field ground state of the t-J model. It leads to the stripe patterns we have found numerically in a generalized t-J-type model including mass renormalization from the electron-phonon coupling. In the hole-doped side, a robust d-wave superconducting order results in the formation of the half-doped antiferromagnetic resonating-valence-bond (AF-RVB) stripes shown by the well-known Yamada plot. On the other hand, due to a long-range AF order in electron-doped materials, a stripe structure with the "in-phase" magnetic domain (IPMD) is obtained in the underdoped regime instead of the AF-RVB stripe. The IPMD stripe with the largest period permitted by lattice size is stabilized near the underdoped region and it excludes the Yamada plot from electron-doped cases. Based on finite lattice size to which we can reach, the existence of IPMD stripes may imply an electronic phase separation into an electron-rich and an insulating half-filled AF long-range ordered domains in electron-doped compounds.
Motivated by numerical evidence of the valence-bond ground state of the two-dimensional Heisenberg pyrochlore lattice, we argue using a t-J model that it evolves under doping into novel phases characterized by superconductivity coexisting with the underlying valence-bond solid order. A fermionic mean-field theory supplemented by exact diagonalization results provide strong arguments in favor of the stability of such supersolid phases. The resemblance with modulated superconducting patterns in high-Tc cuprates as well as possible relevance to frustrated noncuprate superconductors such as spinels and pyrochlores is discussed.