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 study a model in (2 + 1)-dimensional space-time that is realized by an array of chains, each of which realizes relativistic Majorana fields in (1 + 1)-dimensional space-time, coupled via current-current interactions. The model is shown to have a lattice realization as an array of coupled quantum spin-1/2 ladders. We study this model both in the presence and in the absence of time-reversal symmetry within a mean-field approximation. We find regimes in coupling space where Abelian and non-Abelian spin-liquid phases are stable. In the case when the Hamiltonian is time-reversal symmetric, we find regimes where gapped Abelian and non-Abelian chiral phases appear as a result of spontaneous breaking of time-reversal symmetry. These gapped phases are separated by a discontinuous phase transition. More interestingly, we find a regime for which a nonchiral gapless non-Abelian spin liquid is stable. The excitations in this regime are described by relativistic Majorana fields in (2 + 1)-dimensional space-time, much as those appearing in the Kitaev honeycomb model but here emerging in a model of coupled spin ladders that do not break the
SU(2) spin-rotation symmetry.
In the physics of the fractional quantum Hall (FQH) effect, a zoo of Abelian topological phases can be obtained by varying the magnetic field. Aiming to reach the same phenomenology in spin like systems, we propose a family of
SU(N)-symmetric models in the fundamental representation, on the square lattice with short-range interactions restricted to triangular units, a natural generalization for arbitrary
N of an SU(3) model studied previously where time-reversal symmetry is broken explicitly. Guided by the recent discovery of
SU(2)1 and
SU(3)1 chiral spin liquids (CSL) on similar models we search for topological
SU(N)1 CSL in some range of the Hamiltonian parameters via a combination of complementary numerical methods such as exact diagonalizations (ED), infinite density matrix renormalization group (iDMRG) and infinite Projected Entangled Pair State (iPEPS). Extensive ED on small (periodic and open) clusters up to
N=10 and an innovative
SU(N)-symmetric version of iDMRG to compute entanglement spectra on (infinitely long) cylinders in all topological sectors provide unambiguous signatures of the
SU(N)1 character of the chiral liquids. An SU(4)-symmetric chiral PEPS, constructed in a manner similar to its SU(2) and SU(3) analogs, is shown to give a good variational ansatz of the
N=4 ground state, with chiral edge modes originating from the PEPS holographic bulk-edge correspondence. Finally, we discuss the possible observation of such Abelian CSL in ultracold atom setups where the possibility of varying
N provides a tuning parameter similar to the magnetic field in the physics of the FQH effect.
Abelian and non-Abelian topological phases exhibiting protected chiral edge modes are ubiquitous in the realm of the fractional quantum Hall (FQH) effect. Here, we investigate a spin-1 Hamiltonian on the square lattice which could, potentially, host the spin liquid analog of the (bosonic) non-Abelian Moore-Read FQH state, as suggested by exact diagonalization of small clusters. Using families of fully SU(2)-spin symmetric and translationally invariant chiral projected entangled pair states (PEPS), variational energy optimization is performed using infinite-PEPS methods, providing good agreement with density matrix renormalization group (DMRG) results. A careful analysis of the bulk spin-spin and dimer-dimer correlation functions in the optimized spin liquid suggests that they exhibit long-range ``gossamer tails''. From the investigation of the entanglement spectrum, we observe sharply defined chiral edge modes following the prediction of the SU(2)
2 Wess-Zumino-Witten theory and exhibiting a conformal field theory (CFT) central charge
c=3/2, as expected for a Moore-Read chiral spin liquid. Using the PEPS bulk-edge correspondence, we argue the ``weak'' criticality of the bulk is in fact a finite-
D artifact of the chiral PEPS, which quickly becomes (practically) irrelevant as the PEPS bond dimension
D is increased. We conclude that the PEPS formalism offers an unbiased and efficient method to investigate non-Abelian chiral spin liquids in quantum antiferromagnets.
We study the interplay of competing interactions in spin-
21 triangular Heisenberg model through tuning the first- (
J1), second- (
J2), and third-neighbor (
J3) couplings. Based on a large-scale density-matrix renormalization group calculation, we identify a quantum phase diagram of the system and discover a gapless chiral spin-liquid (CSL) phase in the intermediate
J2 and
J3 regime. This CSL state spontaneously breaks time-reversal symmetry with finite scalar chiral order, and it has gapless excitations implied by a vanishing spin triplet gap and a finite central charge on the cylinder. Moreover, the central charge grows rapidly with the cylinder circumference, indicating emergent spinon Fermi surfaces. To understand the numerical results we propose a parton mean-field spin-liquid state, the
U(1) staggered flux state, which breaks time-reversal symmetry with chiral edge modes by adding a Chern insulator mass to Dirac spinons in the
U(1) Dirac spin liquid. This state also breaks lattice rotational symmetries and possesses two spinon Fermi surfaces driven by nonzero
J2 and
J3, which naturally explains the numerical results. This realizes an example of a gapless CSL state with coexisting spinon Fermi surfaces and chiral edge states, demonstrating the rich family of interesting quantum phases emergent from competing interactions in triangular-lattice magnets.
We describe a coupled-chain construction for chiral spin liquids in two-dimensional spin systems. Starting from a one-dimensional zigzag spin chain and imposing SU(2) symmetry in the framework of non-Abelian bosonization, we first show that our approach faithfully describes the low-energy physics of an exactly solvable model with a three-spin interaction. Generalizing the construction to the two-dimensional case, we obtain a theory that incorporates the universal properties of the chiral spin liquid predicted by Kalmeyer and Laughlin: charge-neutral edge states, gapped spin-1/2 bulk excitations, and ground-state degeneracy on the torus signaling the topological order of this quantum state. In addition, we show that the chiral spin liquid phase is more easily stabilized in frustrated lattices containing corner-sharing triangles, such as the extended kagome lattice, than in the triangular lattice. Our field-theoretical approach invites generalizations to more exotic chiral spin liquids and may be used to assess the existence of the chiral spin liquid as the ground state of specific lattice systems.