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Journal of Physics Condensed Matter

Staggered flux phase and d-wave phase of the t-J model

Guang-Jie Chen, Robert Joynt, Fu‐Chun Zhang

Published Jul 8, 1991• 4 citations
Publisher

Abstract

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.

DOI
10.1088/0953-8984/3/27/016

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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.
Starting from the large-U Hubbard model combining the intersite Coulomb interaction V, we investigate the competition, on a square lattice, between the d-wave resonating-valence-bond (RVB) state and the staggered-flux-phases (SFP) state on the basis of F. C. Zhang's earlier work [Phys. Rev. Lett. 64, 974 (1990)]. We find that a very small V (0.1U) may help stabilize the SFP state by effectively suppressing the d-wave pairing amplitude and enhance the particle-hole correlation. We show in the phase diagram of V-\ensuremath{\delta} (the doping concentration) the critical curve of Vc{\mathit{V}}_{\mathit{c}}Vc​(\ensuremath{\delta}) below which the RVB state is energetically favored while beyond this point a mixed state of SFP and RVB occurs, which has the lowest energy. We also find that the pure SFP state is not the most stable state in the whole V region.
We study the stability of the Fermi-liquid and staggered flux phases in the t-J model for densities off half-filling using a large-N slave-boson approach. We examine the stability of the Fermi-liquid phase with respect to flux-density-wave (FDW) and bond-density-wave (BDW), or spin-Peierls, fluctuations at arbitrary wave vector as a function of doping \ensuremath{\delta} and the ratio \ensuremath{\delta}t/J. For densities near half-filling the Fermi liquid becomes unstable below a critical tc{\mathit{t}}_{\mathit{c}}tc​ towards an FDW state with an incommensurate wave vector k near (\ensuremath{\pi},\ensuremath{\pi}). The critical ratio \ensuremath{\delta}tc{\mathit{t}}_{\mathit{c}}tc​/J for the Fermi-liquid--FDW transition diverges rapidly as \ensuremath{\delta}\ensuremath{\rightarrow}0. For \ensuremath{\delta}>18%, the FDW instability of the Fermi liquid is preempted by a BDW instability at incommensurate k. We examine the stability of the \ensuremath{\surd}2 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}2 staggered flux phase with respect to coupled FDW and BDW fluctuations and find that, upon lowering t/J at small \ensuremath{\delta}, the staggered flux phase remains stable for \ensuremath{\delta}t/J above a nonzero critical ratio and then develops an instability of primarily BDW character. At nonzero \ensuremath{\delta} and still lower \ensuremath{\delta}t/J, the staggered flux phase also develops an instability with respect to long-wavelength FDW fluctuations; tc{\mathit{t}}_{\mathit{c}}tc​/J for this instability is approximately independent of \ensuremath{\delta}. We study the response function for such fluctuations analytically, showing the nontrivial k dependence induced by the presence of a small density of holes in the staggered phase. It is conjectured that this long-wavelength FDW instability is associated with the existence off half-filling of a commensurate flux phase with lower energy than the staggered flux phase.
We study the stability of the staggered-flux phase in the t-J model away from half-filling using a systematic large-N slave-boson approach. Found below a critical doping concentration ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$=a(J/t)2{)}^{2})2, is a flux-density-wave instability with an incommensurability wave number \ensuremath{\sim}${\mathrm{\ensuremath{\delta}}}^{3/4}$. The instability towards a modulated-flux state is due to low-lying phase fluctuations of the valence bonds. When the doping parameter exceeds ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$, we find a fully gapped d-wave superconducting state due to residual quasiparticle interactions.
We have numerically evaluated the energy of several kinds of wave functions considered to be candidate ground states of the two-dimensional t-J model at various hole densities. We searched a parameter space which includes d-wave and s-wave superconductivity and spin-density-wave ordering as well as the projected Fermi-liquid state. Coexistence of different orderings, such as the s+id state and d-wave spin-density-wave state, were found to be stable states. We find a phase diagram in the density-t/J plane which has coexistence of antiferromagnetism and superconductivity at very low hole concentrations and superconductivity up to rather high values of density---about 40%. At intermediate concentrations, the time-reversal symmetry-breaking s+id state is found.