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 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.
We study staggered flux fluctuations around the superconducting state of the SU(2) mean-field theory for the two-dimensional t-J model and their effect on the electron spectral function. The quasiparticle peaks near (pi,0),(0,pi) get strongly broadened and partially wiped out by these fluctuations while the quasiparticle peaks near the nodes of the d-wave gap are preserved over a wide parameter range. The strength of these effects is governed by an energy scale that decreases towards zero for doping x-->0 and that is related to the energy splitting between the SU(2)-related superconducting and staggered flux mean-field states.
We study the relevance of the staggered-flux phase in the $t\ensuremath{-}J$ model using a system with two holes on a 32-site lattice with periodic boundary conditions. We find a staggered-flux pattern in the current-current correlation in the lowest energy d-wave state where there is mutual attraction between the holes. This staggered correlation decays faster with distance when the hole binding becomes stronger. This is in complete agreement with a recent study by Ivanov, Lee, and Wen [Phys. Rev. Lett. 84, 3958 (2000)] based on the SU(2) theory, and strongly suggests that the staggered-flux phase is a key ingredient in the $t\ensuremath{-}J$ model. We further show that this staggered-flux pattern does not exist in a state where the holes repel each other. Correlations of the chirality operator ${\mathbf{S}}_{1}\ensuremath{\cdot}({\mathbf{S}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbf{S}}_{3})$ show that the staggered pattern of the chirality is closely tied to the holes.
The ground state of the bipartite $t\text{\ensuremath{-}}J$ model must satisfy a specific sign structure, based on which the single-hole and two-hole ground state $Ans\stackrel{\ifmmode \ddot{}\else \"{}\fi{}}{a}tze$ on honeycomb lattice are constructed and studied by a variational Monte Carlo (VMC) method. The VMC results are in good agreement with the exact diagonalization (ED) calculation. For the single-hole case, the degenerate ground states are characterized by quantum numbers of a spin-1/2 and an orbital angular momentum ${L}_{z}=\ifmmode\pm\else\textpm\fi{}2$. The latter is associated with the emergent chiral spin/hole currents mutually surrounding the hole/spin-1/2 as a composite object or ``twisted hole''. A vanishing quasiparticle spectral weight is shown in the large-sample limit. In the two-hole ground state, the holes form a spin-singlet pairing with
d+id symmetry in the Cooper channel, but are of
s-wave symmetry as a tightly bound pair of the ``twisted holes''. Such a pairing mechanism of dichotomy can be attributed to eliminating the local spin currents which has nothing to do with the long-range antiferromagnetic correlation. Superconducting ground state at finite doping is briefly discussed in terms of the tightly bound hole pairs as the building blocks.
With cuprate superconductors in mind, we check whether the d -wave superconducting ( d SC) state coexists with or excludes the staggered flux (SF) state, which was revealed to be a strong candidate for the pseudogap state. To this end, we use a variational Monte Carlo method for the square-lattice t - J model with diagonal transfer t '. We construct a trial wave function which can simultaneously represent d SC and SF orders for a continuous description of their interplay. Furthermore, a band renormalization effect is introduced into both orders independently. It is found that the SF state does not coexist with d SC and is unstable for a wide area in the parameter space. However, a strange state having both SF and antiferromagnetic orders is found to be stable for very low doping rate.