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arXiv:1105.4348 · Physical Review B

Non-Abelian statistics and a hierarchy of fractional spin liquids in spin- S antiferromagnets

Burkhard Scharfenberger, Ronny Thomale, Martin Greiter

Published Oct 4, 2011• 33 citations
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Abstract

We propose a hierarchy of spin liquids with spin $S$, which are generated via a Schwinger boson projection from $2S$ (Abelian) chiral spin liquids with $S=1/2$. We argue that the P and T invariant liquids we construct for integer spin $S$ support SU(2) level $k=S$ non-Abelian statistics. We conjecture that these spin liquids serve as paradigms for antiferromagnets which are disordered through mobile charge carriers.

DOI
10.1103/physrevb.84.140404

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We propose a hierarchy of spin liquids with spin SSS, which are generated via a Schwinger boson projection from 2S2S2S (Abelian) chiral spin liquids with S=1/2S=1/2S=1/2. We argue that the P and T invariant liquids we construct for integer spin SSS support SU(2) level k=Sk=Sk=S non-Abelian statistics. We conjecture that these spin liquids serve as paradigms for antiferromagnets which are disordered through mobile charge carriers.
We study a strongly correlated spin-1 Bose gas in 2D space by using the projective construction. A spin-disordered superfluid state is constructed and proposed as a candidate competing with the conventional polar condensate when interaction is antiferromagnetic. This novel state has a non-trivial topological order whose low energy excitations carry fractional spin, charge, and statistics. The spin excitations become gapless only at the edge and are described by level-1 SU(2) Kac-Moody algebra. The edge state is identical to the edge state of the chiral spin liquid or the right moving sector of spin-1/2 chain.
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{}_{2}2​ Wess-Zumino-Witten theory and exhibiting a conformal field theory (CFT) central charge c=3/2c=3/2c=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-DDD artifact of the chiral PEPS, which quickly becomes (practically) irrelevant as the PEPS bond dimension DDD 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 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{}_{n}n​ Wess-Zumino-Novikov-Witten conformal field theory. The models are constructed from an array of generalized spin-n/2n/2n/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{}_{n}n​ gapless edge excitations.
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-111 square-lattice model with the first- and second-neighbor bilinear (J1,J2J_1,J_2J1​,J2​) and biquadratic interactions (K1,K2K_1,K_2K1​,K2​), as well as the three-spin scalar chiral coupling JχJ_χJχ​. Inspired by the evidence of the Moore-Read state in exact diagonalization and infinite Projected Entangled Pair States calculations, we fix J1=1.0J_1=1.0J1​=1.0, J2/J1=0.623J_2/J_1=0.623J2​/J1​=0.623, K1/J1=−0.176K_1/J_1=-0.176K1​/J1​=−0.176, and obtain a quantum phase diagram for 0≤K2/J1≤0.40 \leq K_2/J_1 \leq 0.40≤K2​/J1​≤0.4 and 0.3≤Jχ/J1≤0.60.3 \leq J_χ/J_1 \leq 0.60.3≤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=1C=1C=1, and the level counting of low-lying entanglement spectra which is consistent with the chiral SU(2)2_22​ conformal field theory. Our results may shed new light on searching for non-Abelian CSLs in other higher-spin (S>1/2S > 1/2S>1/2) systems with short-range couplings.
We study S=1S=1S=1 spin liquid states on the kagome lattice constructed by Gutzwiller-projected px+ipy{p}_{x}+i{p}_{y}px​+ipy​ superconductors. We show that the obtained spin liquids are either non-Abelian or Abelian topological phases, depending on the topology of the fermionic mean-field state. By calculating the modular matrices SSS and TTT, we confirm that projected topological superconductors are non-Abelian chiral spin liquid (NACSL). The chiral central charge and the spin Hall conductance we obtained agree very well with the SO(3)1SO{(3)}_{1}SO(3)1​ (or, equivalently, SU(2)2SU{(2)}_{2}SU(2)2​) field-theory predictions. We propose a local Hamiltonian which may stabilize the NACSL. From a variational study, we observe a topological phase transition from the NACSL to the Z2{Z}_{2}Z2​ Abelian spin liquid.