By using the variational Monte Carlo technique, we study the spin-
21 XXZ antiferromagnetic model (with easy-plane anisotropy) on the kagome lattice. A class of Gutzwiller projected fermionic states with a spin Jastrow factor is considered to describe either spin liquids [with
U(1) or
Z2 symmetry] or magnetically ordered phases [with
q=(0,0) or $\mathbf{q}=(4\ensuremath{\pi}/3,0)$]. We find that the magnetic states are not stable in the thermodynamic limit. Moreover, there is no energy gain to break the gauge symmetry from
U(1) to
Z2 within the spin-liquid states, as previously found in the Heisenberg model. The best variational wave function is therefore the
U(1) Dirac state, supplemented by the spin Jastrow factor. Furthermore, a vanishing
S=2 spin gap is obtained at the variational level, in the whole regime from the
XY to the Heisenberg model.
By using Gutzwiller projected fermionic wave functions and variational Monte Carlo technique, we study the spin-
1/2 Heisenberg model with the first-neighbor (
J1), second-neighbor (
J2), and additional scalar chiral interaction ${J}_{\ensuremath{\chi}}{\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}({\mathbf{S}}_{j}\ifmmode\times\else\texttimes\fi{}{\mathbf{S}}_{k})$ on the triangular lattice. In the nonmagnetic phase of the ${J}_{1}\ensuremath{-}{J}_{2}$ triangular model with $0.08\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.16$, recent density-matrix renormalization group (DMRG) studies [Zhu and White, Phys. Rev. B 92, 041105(R) (2015) and Hu, Gong, Zhu, and Sheng, Phys. Rev. B 92, 140403(R) (2015)] find a possible gapped spin liquid with the signal of a competition between a chiral and a
Z2 spin liquid. Motivated by the DMRG results, we consider the chiral interaction ${J}_{\ensuremath{\chi}}{\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}({\mathbf{S}}_{j}\ifmmode\times\else\texttimes\fi{}{\mathbf{S}}_{k})$ as a perturbation for this nonmagnetic phase. We find that with growing ${J}_{\ensuremath{\chi}}$, the gapless U(1) Dirac spin liquid, which has the best variational energy for ${J}_{\ensuremath{\chi}}=0$, exhibits the energy instability towards a gapped spin liquid with nontrivial magnetic fluxes and nonzero chiral order. We calculate topological Chern number and ground-state degeneracy, both of which identify this flux state as the chiral spin liquid with fractionalized Chern number
C=1/2 and twofold topological degeneracy. Our results indicate a positive direction to stabilize a chiral spin liquid near the nonmagnetic phase of the ${J}_{1}\ensuremath{-}{J}_{2}$ triangular model.
We perform an exact-diagonalization study of the spin-
21 XXZ Heisenberg antiferromagnet on the kagome lattice at finite magnetization
m=32 with an emphasis on the
XY point
(Jz=0) and in the presence of a small chiral term. Recent analytic work by Kumar et al. [K. Kumar, K. Sun, and E. Fradkin, Phys. Rev. B 90, 174409 (2014)] on the same model, using a newly developed flux attachment transformation, predicts a plateau at this value of the magnetization described by a chiral spin liquid (CSL) with a spin Hall conductance of ${\ensuremath{\sigma}}_{xy}=\frac{1}{2}$. Such a state is topological in nature, has a ground-state degeneracy, and exhibits fractional excitations. We analyze the degeneracy structure in the low-energy manifold, identify the candidate topological states, and use them to compute the modular matrices and Chern numbers, all of which strongly agree with expected theoretical behavior for the ${\ensuremath{\sigma}}_{xy}=\frac{1}{2}$ CSL. In the limit of zero chirality, we find on most (not all) clusters that the topological invariants are still those of a CSL.
Motivated by recent density-matrix renormalization group (DMRG) calculations [Yan, Huse, and White, Science 332, 1173 (2011)], which claimed that the ground state of the nearest-neighbor spin-
1/2 Heisenberg antiferromagnet on the kagome lattice geometry is a fully gapped spin liquid with numerical signatures of
Z2 gauge structure, and a further theoretical work [Lu, Ran, and Lee, Phys. Rev. B 83, 224413 (2011)], which gave a classification of all Schwinger-fermion mean-field fully symmetric
Z2 spin liquids on the kagome lattice, we have thoroughly studied Gutzwiller-projected fermionic wave functions by using quantum variational Monte Carlo techniques, hence implementing exactly the constraint of one fermion per site. In particular, we investigated the energetics of all
Z2 candidates (gapped and gapless) that lie in the neighborhood of the energetically competitive U(1) gapless spin liquids. By using a state-of-the-art optimization method, we were able to conclusively show that the U(1) Dirac state is remarkably stable with respect to all
Z2 spin liquids in its neighborhood, and in particular for opening a gap toward the so-called ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state, which was conjectured to describe the ground state obtained by the DMRG method. Finally, we also considered the addition of a small second nearest-neighbor exchange coupling of both antiferromagnetic and ferromagnetic type, and obtained similar results, namely, a U(1) Dirac spin-liquid ground state.
We study the energy and the static spin structure factor of the ground state of the spin-
1/2 quantum Heisenberg antiferromagnetic model on the kagome lattice. By the iterative application of a few Lanczos steps on accurate projected fermionic wave functions and the Green's function Monte Carlo technique, we find that a gapless (algebraic)
U(1) Dirac spin liquid is competitive with previously proposed gapped (topological)
Z2 spin liquids. By performing a finite-size extrapolation of the ground-state energy, we obtain an energy per site $E/J=\ensuremath{-}0.4365(2)$, which is equal, within three error bars, to the estimates given by the density-matrix renormalization group (DMRG). Our estimate is obtained for a translationally invariant system, and, therefore, does not suffer from boundary effects, like in DMRG. Moreover, on finite toric clusters at the pure variational level, our energies are lower compared to those from DMRG calculations.
We investigate the spin-
21 antiferromagnetic Heisenberg model with a Dzyaloshinskii-Moriya interaction on kagome lattice, making use of the variational Monte Carlo technique. An exotic quantum spin state is found to arise from a melting of the $\mathbit{Q}=0$ long-range magnetic order by a topological transition, when a small anisotropic third-nearest-neighbor antiferromagnetic Heisenberg interaction is turned on. This state is a gapped quantum spin liquid, characterized by a topological order with ground-state degeneracy
ng=4 and topological entanglement entropy $\ensuremath{\gamma}=ln2$, suggesting it is an Abelian topological phase. Furthermore, the Chern numbers of the spin-up (-down) spinon occupied bands of this state are ${C}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}=\ifmmode\pm\else\textpm\fi{}1$, respectively. From this perspective, this state is also a time-reversal symmetric (total Chern number
Ctotal=0) topological insulator with spinons as the chiral edge states, which carry opposite spin and move in the opposite direction. It is analogous to the quantum spin Hall state but the spin current is carried by deconfined spinons in a quantum spin liquid, and so is dubbed as the spinon quantum spin Hall state.