We construct a two-dimensional (2D) lattice model that is argued to realize a gapped chiral spin liquid with (Ising) non-Abelian topological order. The building blocks are spin-
21 two-leg ladders with
SU(2)-symmetric spin-spin interactions. The two-leg ladders are then arranged on rows and coupled through
SU(2)-symmetric interactions between consecutive ladders. The intraladder interactions are tuned so as to realize
c=1/2 Ising conformal field theory, a fact that we establish numerically via density matrix renormalization group studies. Time-reversal breaking interladder interactions are tuned so as to open a bulk gap in the 2D lattice system. This 2D system supports gapless chiral edge modes with Ising non-Abelian excitations but no charge excitations, in contrast to the Pfaffian non-Abelian fractional quantum Hall state.
We suggest a class of two-dimensional lattice spin Hamiltonians describing non-Abelian SU(2) chiral spin liquids---spin analogs of fractional non-Abelian quantum Hall states---with gapped bulk and gapless chiral edge excitations described by the SU(2)
n Wess-Zumino-Novikov-Witten conformal field theory. The models are constructed from an array of generalized spin-
n/2 ladders with multi-spin-exchange interactions which are coupled by isolated spins. Such models allow a controllable analytic treatment starting from the one-dimensional limit and are characterized by a bulk gap and non-Abelian SU(2)
n gapless edge excitations.
We revisit the fermionic parton approach to
S=1/2 quantum spin liquids with
SU(2) spin-rotation symmetry, and the associated projective symmetry group (PSG) classification. We point out that the existing PSG classification is incomplete; upon completing it, we find spin-liquid states with
S=1 and
S=0 Majorana fermion excitations coupled to a deconfined
Z2 gauge field. The crucial observation leading us to this result is that, like space group and time-reversal symmetries, spin rotations can act projectively on the fermionic partons; that is, a spin rotation may be realized by simultaneous
SU(2) spin and gauge rotations. We show that there are only two realizations of spin rotations acting on fermionic partons: the familiar naive realization where spin rotation is not accompanied by any gauge transformation, and a single type of projective realization. We discuss the PSG classification for states with projective spin rotations. To illustrate these results, we show that there are four such PSGs on the two-dimensional square lattice. We study the properties of the corresponding states, finding that one---with gapless Fermi points---is a stable phase beyond mean-field theory. In this phase, depending on parameters, a small Zeeman magnetic field can open a partial gap for the Majorana fermion excitations. Moreover, there are nearby gapped phases supporting
Z2 vortex excitations obeying non-Abelian statistics. We conclude with a discussion of various open issues, including the challenging question of where such
S=1 Majorana spin liquids may occur in models and in real systems.
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 describe a new class of spin liquids with global SU(2) spin-rotation symmetry in spin-1/2 systems on the triangular lattice, which have real Majorana fermion excitations carrying spin
S=1. The simplest translationally invariant mean-field state on the triangular lattice breaks time-reversal symmetry and is stable to fluctuations. It generically possesses gapless excitations along three Fermi lines in the Brillouin zone. These intersect at a single point where the excitations scale with a dynamic exponent
z=3. An external magnetic field has no orbital coupling to the SU(2) spin-rotation-invariant fermion bilinears that can give rise to a transverse thermal conductivity, thus leading to the absence of a thermal Hall effect. The Zeeman coupling is found to gap out two-thirds of the
z=3 excitations near the intersection point and this leads to a suppression of the low-temperature specific heat, the spin susceptibility, and the Wilson ratio. We also compute physical properties in the presence of weak disorder and discuss possible connections to recent experiments on organic insulators.
We study an exactly solvable model with bond-directional quadrupolar and octupolar interactions between spin-orbital entangled
jeff=23 moments on the honeycomb lattice. We show that this model features a multipolar spin liquid phase with gapless fermionic excitations. In the presence of perturbations that break time-reversal and rotation symmetries, we find Abelian and non-Abelian topological phases in which the Chern number evaluates to 0, $\ifmmode\pm\else\textpm\fi{}1$, and $\ifmmode\pm\else\textpm\fi{}2$. We also investigate quantum phase transitions out of the multipolar spin liquid using a parton mean-field approach and orbital wave theory. In the regime of strong integrability-breaking interactions, the multipolar spin liquid gives way to ferroquadrupolar-vortex and antiferro-octupolar ordered phases that harbor a hidden spin-
21 Kitaev spin liquid. Our work unveils mechanisms for unusual multipolar orders and quantum spin liquids in Mott insulators with strong spin-orbit coupling.