We introduce an exactly solvable SU(2)-invariant spin-1/2 model with exotic spin excitations. With time reversal symmetry (TRS), the ground state is a spin liquid with gapless or gapped spin-1 but fermionic excitations. When TRS is broken, the resulting spin liquid exhibits deconfined vortex excitations which carry spin-1/2 and obey non-Abelian statistics. We show that this SU(2) invariant non-Abelian spin liquid exhibits the spin quantum Hall effect with quantized spin Hall conductivity σ(xy)(s)=ℏ/2π, and that the spin response is effectively described by the SO(3) level-1 Chern-Simons theory at low energy. We further propose that a SU(2) level-2 Chern-Simons theory is the effective field theory describing the topological structure of the non-Abelian SU(2) invariant spin liquid.
We study the spin-1/2 Heisenberg model on the triangular lattice with the antiferromagnetic first (J1) and second nearest-neighbor (J2) interactions using density matrix renormalization group. By studying the spin correlation function, we find a 120◦ magnetic order phase for J2 . 0.08J1 and a stripe antiferromagnetic phase for J2 & 0.16J2. Between these two phases, we identify a gapped spin liquid phase characterized by the exponential decaying spin and dimer correlations, as well as finite spin singlet and triplet excitation gaps. We also find two topological degenerating ground states in two topological sectors. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. While the sector with spinon is found to respect the time reveral symmetry, the even sector without a spinon may break such a symmetry. We discuss the implications of numerical results for the nature of the topological spin liquid phase.
We establish the existence of a chiral spin liquid (CSL) as the exact ground state of the Kitaev model on a decorated honeycomb lattice, which is obtained by replacing each site in the familiar honeycomb lattice with a triangle. This CSL state spontaneously breaks time reversal symmetry but preserves other symmetries. There are two topologically distinct CSL's separated by a quantum critical point. Interestingly, vortex excitations in the topologically nontrivial (Chern number +/-1) CSL obey non-Abelian statistics.
We study the zero-temperature phase diagram of a class of two-dimensional SU
(N) antiferromagnets. These models are characterized by having the same type of SU
(N) spin placed at each site of the lattice, and share the property that, in general, more than two spins must be combined to form a singlet. An important motivation to study these systems is that they may be realized naturally in Mott insulators of alkaline-earth atoms placed on optical lattices; indeed, such Mott insulators have already been obtained experimentally, although the temperatures are still high compared to the magnetic exchange energy. We study these antiferromagnets in a large-
N limit, finding a variety of ground states. Some of the models studied here have a valence-bond solid ground state, as was found in prior studies, yet we find that many others have a richer variety of ground states. Focusing on the two-dimensional square lattice, in addition to valence cluster states (which are analogous to valence-bond states), we find both Abelian and non-Abelian chiral spin liquid ground states, which are magnetic counterparts of the fractional quantum Hall effect. We also find a ``doubled'' chiral spin liquid ground state that preserves time-reversal symmetry. These results are based on a combination of rigorous lower bounds on the large-
N ground-state energy and a systematic numerical ground-state search. We conclude by discussing whether experimentally relevant SU
(N) antiferromagnets, away from the large-
N limit, may be chiral spin liquids.
Non-Abelian chiral spin liquids (CSLs) host non-Abelian anyon excitations and are promising platforms for topological quantum computation. However, realizing non-Abelian CSLs in short-range interacting spin systems remains elusive. Here, we study a spin-
1 square-lattice model with the first- and second-neighbor bilinear (
J1,J2) and biquadratic interactions (
K1,K2), as well as the three-spin scalar chiral coupling
Jχ. Inspired by the evidence of the Moore-Read state in exact diagonalization and infinite Projected Entangled Pair States calculations, we fix
J1=1.0,
J2/J1=0.623,
K1/J1=−0.176, and obtain a quantum phase diagram for
0≤K2/J1≤0.4 and
0.3≤Jχ/J1≤0.6 by using the density matrix renormalization group (DMRG) calculations. We identify a stripe antiferromagnetic phase, a Néel antiferromagnetic phase, a magnetically ordered chiral spin state phase, and a non-Abelian CSL phase emerging near the boundaries of the ordered phases. The unbiased DMRG results provide strong evidence for the Moore-Read state, including the three topological sectors, the quantized spin Chern number
C=1, and the level counting of low-lying entanglement spectra which is consistent with the chiral SU(2)
2 conformal field theory. Our results may shed new light on searching for non-Abelian CSLs in other higher-spin (
S>1/2) systems with short-range couplings.