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arXiv:cond-mat/0007068 · Physical review. B, Condensed matter

Signatures of a staggered-flux phase in the t−J model with two holes on a 32-site lattice

P. W. Leung

Published Sep 1, 2000• 54 citations
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Abstract

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.

DOI
10.1103/physrevb.62.r6112

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We study the relevance of the staggered-flux phase in the t\ensuremath−Jt\ensuremath{-}Jt\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−Jt\ensuremath{-}Jt\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 S1\ensuremath⋅(S2\ifmmode×\else\texttimes\fiS3){\mathbf{S}}_{1}\ensuremath{\cdot}({\mathbf{S}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbf{S}}_{3})S1​\ensuremath⋅(S2​\ifmmode×\else\texttimes\fiS3​) show that the staggered pattern of the chirality is closely tied to the holes.
We study the ``t-J'' model with strongly correlated electrons (infinite-U Hubbard-model kinetic energy plus superexchange J) by exact diagonalization on small two-dimensional clusters of six, eight, and twelve sites on a square lattice with periodic boundary conditions. We find an attractive pairing potential for two holes when J/t\ensuremath{\ge}(J/t)c{)}_{\mathit{c}})c​\ensuremath{\simeq}0.2, with t the hopping matrix element. Ground-state calculations show that extended s-wave and d-wave pairing susceptibilities and equal-time pairing correlations grow weakly with increasing J and system size at fixed filling n. Pairing correlations may be enhanced for (J/t)>(J/t)c{)}_{\mathit{c}})c​\ensuremath{\approxeq}0.2. We compare equal-time pairing correlations to a mean-field theory for J=0 and note potential difficulties with interpreting numerical results for such small systems. The widths of one- and two-hole energy bands in low-spin states are of order J; the one-hole effective mass is larger than the two-hole effective mass in such states. Spin-charge correlation functions for a single hole in the eight- and twelve-site lattices are consistent with the spin-charge decoupling scenario for resonating-valence-bond states.
Antiferromagnetic flux states for the t-J model are investigated on a finite 4\ifmmode\times\else\texttimes\fi{}4 cluster with 2 holes. These wave functions consist of projected Slater determinants of noninteracting wave functions of electrons moving on a lattice with alternating positive and negative magnetic fluxes flowing through the two sublattices of unit cells. Following a recent approach used for commensurate flux phases, we also evaluate the exchange energy and the kinetic energy in the thermodynamic limit. At small J/t ratio the commensurate flux phase is shown to be unstable against the metallic staggered flux phase.
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.
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.
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.