Motivated by recent experiments on the material Ba
3NiSb
2O
9, we consider a spin-one quantum antiferromagnet on a triangular lattice with the Heisenberg bilinear and biquadratic exchange interactions and a single-ion anisotropy. Using a fermionic ``triplon'' representation for spins, we study the phase diagram within mean-field theory. In addition to a fully gapped spin-liquid ground state, we find a state where one gapless triplon mode with a Fermi surface coexists with
d+id topological pairing of the other triplons. Despite the existence of a Fermi surface, this ground state has fully gapped bulk spin excitations. Such a state has linear in-temperature specific heat and constant in-plane spin susceptibility, with an unusually high Wilson ratio.
We consider a SU(3) ring-exchange model on a triangular lattice. Unlike the SU(2) case, under perturbation expansion of the SU(3) Hubbard model, the three-site ring exchange is present as well as the usual four-site ring exchange. Interestingly, the three-site ring exchange differs from the usual two-site and four-site exchanges by a minus sign and is ferromagnetic. We first present numerical site-factorized state studies on this model which shows a three-sublatticed order phase and a ferromagnetic phase. We further study the model using slave-fermion mean field in which we rewrite the exchange operators in terms of three flavors of fermions. We find the main competing trial states are the trimer state (triangular plaquette state) and the gapless U(1) spin liquid states with parton Fermi surfaces which include both the uniform zero-flux spin liquid state, and the uniform $\ensuremath{\pi}$-flux spin liquid state. Furthermore, we find there are possible pairing instabilities of the zero-flux (Fermi surface) spin liquid state toward a
f-wave gapless (nodal) spin liquid state and the $\ensuremath{\pi}$-flux (Fermi surface) spin liquid state toward an interesting exotic
s-wave ``gapless'' spin liquid state with two flavors of fermions paired up while one flavor of fermions remains gapless.
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.
Motivated by recent experiments on the Heisenberg
S=1/2 quantum spin liquid candidate material kapellasite, we classify all possible chiral (time-reversal symmetry breaking) spin liquids with fermionic spinons on the kagome lattice. We obtain the phase diagram for the physically relevant extended Heisenberg model, comparing the energies of a wide range of microscopic variational wave functions. We propose that, at low temperature, kapellasite exhibits a gapless chiral spin liquid phase with spinon Fermi surfaces. This two-dimensional state inherits many properties of the nearby one-dimensional phase of decoupled antiferromagnetic spin chains, but also shows some remarkable differences. We discuss the spin structure factors and other physical properties.
We investigate the spin-
21 nearest-neighbor Heisenberg model with the four-site ring-exchange
J4 and chiral interaction ${J}_{\ensuremath{\chi}}$ on the triangular lattice by using the variational Monte Carlo method. The
J4 term induces the quadratic band touching (QBT) quantum spin liquid (QSL) with only a
d+id spinon pairing (without hopping term), the nodal
d-wave QSL and
U(1) QSL with a finite spinon Fermi surface progressively. The effect of the chiral interaction ${J}_{\ensuremath{\chi}}$ can enrich the phase diagram with two interesting chiral QSLs (topological orders) with the same quantized Chern number
C=21 and ground-state degeneracy GSD = 2, namely the
U(1) chiral spin liquid (CSL) and
Z2d+id-wave QSL. The nodal
d-wave QSL is fragile and will turn to the
Z2d+id QSL with any finite ${J}_{\ensuremath{\chi}}$ within our numerical calculation. However, in the process from QBT to the
Z2d+id QSL with the increase of ${J}_{\ensuremath{\chi}}$, an exotic crossover region is found. In this region, the previous QBT state acquires a small hopping term so that it opens a small gap at the band touching points, and leads to an energy minimum which is energetically more favorable compared to another competitive local minimum from the
Z2d+id QSL. We dub this state as the proximate QBT QSL and it gives way to the
Z2d+id QSL eventually. Therefore, the cooperation of the
J4 and ${J}_{\ensuremath{\chi}}$ terms favors mostly the
Z2d+id-wave QSL, so that this phase occupies the largest region in the phase diagram.
The possible ground states of the undoped and doped Kitaev–Heisenberg model on a triangular lattice are studied. For the undoped system, a combination of the numerical exact diagonalization calculation and the four-sublattice transformation analysis suggests one possible exotic phase and four magnetically ordered phases, including a collinear stripe pattern and a noncollinear spiral pattern in the global phase diagram. The exotic phase near the antiferromagnetic (AF) Kitaev point is further investigated using the Schwinger-fermion mean-field method, and we obtain an energetically favorable Z 2 chiral spin liquid with a Chern number ±2 as a promising candidate. At finite doping, we find that the AF Heisenberg coupling supports an s -wave or a -wave superconductivity (SC), while the AF and the ferromagnetic Kitaev interactions favor a -wave SC and a time-reversal invariant topological p -wave SC, respectively. Possible experimental realizations and related candidate materials are also discussed.