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arXiv:cond-mat/0601301 · Physical Review B

Bond-order-modulated staggered-flux phase of the tJ model on a square lattice

Cédric Weber, Didier Poilblanc, Sylvain Capponi, Frédéric Mila, Cyril Jaudet

Published Sep 14, 2006• 10 citations
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Abstract

Motivated by the observation of inhomogeneous patterns in some high-${T}_{c}$ 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 $1∕8$ 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 $d$-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.

DOI
10.1103/physrevb.74.104506

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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\ensuremath-J\ensuremath-Vt\text{\ensuremath{-}}J\text{\ensuremath{-}}Vt\ensuremath-J\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×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2, 8\ifmmode×\else\texttimes\fi8\sqrt{8}\ifmmode\times\else\texttimes\fi{}\sqrt{8}8​\ifmmode×\else\texttimes\fi8​, 4\ifmmode×\else\texttimes\fi44\ifmmode\times\else\texttimes\fi{}44\ifmmode×\else\texttimes\fi4, and 32\ifmmode×\else\texttimes\fi32\sqrt{32}\ifmmode\times\else\texttimes\fi{}\sqrt{32}32​\ifmmode×\else\texttimes\fi32​. It is found that the 4\ifmmode×\else\texttimes\fi44\ifmmode\times\else\texttimes\fi{}44\ifmmode×\else\texttimes\fi4 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×\else\texttimes\fi44\ifmmode\times\else\texttimes\fi{}44\ifmmode×\else\texttimes\fi4 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 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 ttt-JJJ model. It leads to the stripe patterns we have found numerically in a generalized ttt-JJJ-type model including mass renormalization from the electron-phonon coupling. In the hole-doped side, a robust ddd-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.
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.
Using Gutzwiller-projected wave functions, I estimate the ground-state energy of the $t\text{\ensuremath{-}}J$ model for several variational states relevant for high-temperature cuprate superconductors. The results indicate that both the superconducting state and the staggered-flux normal state (proposed for vortex cores) are unstable at low doping towards antiferromagnetism and towards phase separation. While phase separation in the underdoped superconducting state may be relevant for the stripe formation mechanism, the results for the normal state suggest that similar charge inhomogeneities may also appear in vortex cores up to relatively high doping values.
Based on the SU(2) lattice-gauge-theory formulation of the $t\ensuremath{-}J$ model, we discuss a possible signature of the unit-cell doubling associated with the staggered-flux (SF) state in the lightly doped spin liquid. Although the SF state appears only dynamically in a uniform d-wave superconducting state, a topological defect [SU(2) vortex] freezes the SF state inside the vortex core. Consequently, the unit-cell doubling shows up in the hopping $({\ensuremath{\chi}}_{\mathrm{ij}})$ and pairing $({\ensuremath{\Delta}}_{\mathrm{ij}})$ order parameters of physical electrons. We find that whereas the center in the vortex core is a SF state, as one moves away from the core center, a correlated staggered modulation of ${\ensuremath{\chi}}_{\mathrm{ij}}$ and ${\ensuremath{\Delta}}_{\mathrm{ij}}$ becomes predominant. We predict that over the region outside the core and inside the internal gauge-field-penetration depth around a vortex center, the local density of states exhibits a staggered peak-dip (SPD) structure inside the V-shaped profile when measured on the bonds. The SPD structure has its direct origin in the unit-cell doubling associated with the SF core and the robust topological texture, which has little to do with the symmetry of the d-wave order parameter. Therefore the structure may survive the tunneling-matrix-element effects and easily be detected by the scanning-tunnel-microscope experiment.
A normal (nonsuperconducting) ground state of the t-J model may be variationally approximated by a Gutzwiller-projected wave function. Within this approximation, at small hole doping near half-filling, the normal state favors staggered-flux ordering. Such a staggered-flux state may occur in vortex cores of underdoped high-temperature cuprate superconductors. From comparing the energies of the staggered-flux state and of the superconducting state, we numerically obtain the condensation energy. Extracting the superfluid density directly from the projected superconducting wave function, we can also estimate the coherence length at zero temperature.
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.