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arXiv:cond-mat/0305143 · Physical review. B, Condensed matter

Staggered-flux normal state in the weakly dopedt-Jmodel

Dmitri A. Ivanov, Patrick A. Lee

Published Oct 6, 2003• 26 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.68.132501

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We develop a slave-boson theory for the $t\ensuremath{-}J$ model at finite doping which respects an $\mathrm{SU}(2)$ symmetry---a symmetry previously known to be important at half filling. The mean field phase diagram is found to be consistent with the phases observed in the cuprate superconductors, which contain $d$-wave superconductor, spin gap, strange metal, and Fermi liquid phases. The spin gap phase is best understood as the staggered flux phase, which is nevertheless translationally invariant for physical quantities. The electron spectral function shows small Fermi pockets at low doping which continuously evolve into the large Fermi surface at high doping concentrations.

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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 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.
Kinetic and magnetic energies of the staggered flux phase, with a fictitious flux of equal magnitude but opposite direction in adjacent square plaquettes, are calculated for the two-dimensional t-J model using the variational Monte Carlo method. They are compared to the energies of the resonating-valence-bond state, the flux phase with half a quantum per plaquette, and the projected Fermi-liquid state. For about 10% hole concentration the staggered flux phase has the lowest energy of the nonsuperconducting states, but its energy is still higher than that of the superconducting d-wave state.
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 the $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ model of a doped Mott insulator in connection to high-Tc{T}_{c}Tc​ superconductors. The nearest neighbor Coulomb interaction (V)(V)(V) is treated quantum mechanically on equal footing as the antiferromagnetic exchange interaction (J)(J)(J). Motivated by the SU(2)SU(2)SU(2) symmetry at half-filling, we construct a large-NNN theory which allows a systematic study of the interplay between staggered flux order and superconductivity upon doping. We solve the model in the large-NNN limit and obtain the ground state properties and the phase diagram as a function of doping. We discuss the competition and the coexistence of the staggered flux and the ddd-wave superconductivity in the underdoped regime and the disappearance of superconductivity in the overdoped regime.
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.
We develop a slave-boson theory for the $t\ensuremath{-}J$ model at finite doping which respects an SU(2)\mathrm{SU}(2)SU(2) symmetry---a symmetry previously known to be important at half filling. The mean field phase diagram is found to be consistent with the phases observed in the cuprate superconductors, which contain ddd-wave superconductor, spin gap, strange metal, and Fermi liquid phases. The spin gap phase is best understood as the staggered flux phase, which is nevertheless translationally invariant for physical quantities. The electron spectral function shows small Fermi pockets at low doping which continuously evolve into the large Fermi surface at high doping concentrations.