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arXiv:1807.04385 · Physical review. B./Physical review. B

Non-Abelian chiral spin liquid in a quantum antiferromagnet revealed by an iPEPS study

Ji-Yao Chen, Laurens Vanderstraeten, Sylvain Capponi, Didier Poilblanc

Published Nov 8, 2018• 84 citations
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Abstract

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.

DOI
10.1103/physrevb.98.184409

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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-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.
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Protected chiral edge modes are a well-known signature of topologically ordered phases, such as the fractional quantum Hall states (FQHS). Using the framework of projected entangled pair states (PEPS) on the square lattice, the authors construct a family of chiral spin-12\frac{1}{2}21​ quantum spin liquids with Z2{\mathbb{Z}}_{2}Z2​ gauge symmetry and analyze in full detail the properties of the edge modes. Surprisingly, the results show that the latter can be well described by a chiral conformal field theory of free bosons (SU(2)1{}_{1}1​), as is the case for the $\ensuremath{\nu}$=12\frac{1}{2}21​ (bosonic) gapped Laughlin state, despite the fact that the numerical data here suggest a critical bulk. The authors thus propose that our family of PEPS physically describes a boundary between a chiral topological phase and a trivial phase and might be closely connected to an (unknown) analogous FQHS.
A simple spin-12\frac{1}{2}21​ frustrated antiferromagnetic Heisenberg model (AFHM) on the square lattice---including chiral plaquette cyclic terms---was argued [A. E. B. Nielsen, G. Sierra, and J. I. Cirac, Nat. Commun. 4, 2864 (2013)] to host a bosonic Kalmeyer-Laughlin (KL) fractional quantum Hall ground state [V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett. 59, 2095 (1987)]. Here, we construct generic families of chiral projected entangled pair states (chiral PEPS) with low bond dimension (D=3,4,5D=3,4,5D=3,4,5) which, upon optimization, provide better variational energies than the KL Ansatz. The optimal D=3D=3D=3 PEPS exhibits chiral edge modes described by the Wess-Zumino-Witten SU(2)1\text{SU}{(2)}_{1}SU(2)1​ model, as expected for the KL spin liquid. However, we find evidence that, in contrast to the KL state, the PEPS spin liquids have power-law dimer-dimer correlations and exhibit a gossamer long-range tail in the spin-spin correlations. We conjecture that these features are genuine to local chiral AFHM on bipartite lattices.
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.
Quantum spin liquids can be faithfully represented and efficiently characterized within the framework of projected entangled pair states (PEPS). Guided by extensive exact diagonalization and density matrix renormalization group calculations, we construct an optimized symmetric PEPS for a SU(3)_{1} chiral spin liquid on the square lattice. Characteristic features are revealed by the entanglement spectrum (ES) on an infinitely long cylinder. In all three Z_{3} sectors, the level counting of the linear dispersing modes is in full agreement with SU(3)_{1} Wess-Zumino-Witten conformal field theory prediction. Special features in the ES are shown to be in correspondence with bulk anyonic correlations, indicating a fine structure in the holographic bulk-edge correspondence. Possible universal properties of topological SU(N)_{k} chiral PEPS are discussed.
Doubts have been raised on the representation of chiral spin liquids exhibiting topological order in terms of projected entangled pair states (PEPSs). Here, starting from a simple spin-1/2 chiral frustrated Heisenberg model, we show that a faithful representation of the chiral spin liquid phase is in fact possible in terms of a generic PEPS upon variational optimization. We find a perfectly chiral gapless edge mode and a rapid decay of correlation functions at short distances consistent with a bulk gap, concomitant with a gossamer long-range tail originating from a PEPS bulk-edge correspondence. For increasing bond dimension, (i) the rapid decrease of spurious features-SU(2) symmetry breaking and long-range tails in correlations-together with (ii) a faster convergence of the ground state energy as compared to state-of-the-art cylinder matrix-product state simulations involving far more variational parameters, prove the fundamental relevance of the PEPS ansatz for simulating systems with chiral topological order.