We compare two time-reversal-symmetric triangular-lattice Hubbard models at half filling: the zero-flux model and a staggered
π-flux model with zero and
π flux through oppositely oriented triangles. Although both reduce to the same Heisenberg model at leading order in the strong-coupling limit, their distinct charge fluctuation profiles yield different quantum paramagnetic regimes. Combining a strong-coupling expansion with infinite density matrix renormalization group (iDMRG) calculations, we find that the positive ring-exchange of the zero-flux model promotes spontaneous chirality, consistent with its notion of a chiral spin liquid at intermediate coupling. In the staggered-
π-flux model, whose noninteracting limit is a Dirac semimetal, the ring exchange changes sign and disfavors chiral order and we find no resolvable scalar chirality. This renders flux-tuned ring-exchange an efficient discriminator between chiral and non-chiral quantum paramagnets at finite coupling.
Chirality and flux correlations are investigated by the exact diagonalization and the quantum Monte Carlo method. In all the cases of the triangular and the square lattice with and without the next nearest-neighbor exchange, the extrapolation to the thermodynamic limit indicates the absence of uniform and staggered chirality in the spin-1/2 Heisenberg models. The spin-1/2 models with the 4-spin and the 6-spin ring exchange show similar behavior to the ordinary Heisenberg models. The quantum Monte Carlo results on the Hubbard model also show no indication of the uniform or the staggered flux order.
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.