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arXiv:1509.00391 · Physical review. B./Physical review. B

Non-Abelian S=1 chiral spin liquid on the kagome lattice

Zheng-Xin Liu, Hong-Hao Tu, Ying-Hai Wu, Rong-Qiang He, Xiong-Jun Liu, Yi Zhou, Tai Kai Ng

Published May 31, 2018• 24 citations
arXivHTMLPDF

Abstract

We study $S=1$ spin liquid states on the kagome lattice constructed by Gutzwiller-projected ${p}_{x}+i{p}_{y}$ 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 $S$ and $T$, 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 $SO{(3)}_{1}$ (or, equivalently, $SU{(2)}_{2}$) 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 ${Z}_{2}$ Abelian spin liquid.

DOI
10.1103/physrevb.97.195158

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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.
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.
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 signalling 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.
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 propose an SU(3) symmetric Hamiltonian with short-range interactions on the kagome lattice and show that it hosts an Abelian chiral spin liquid (CSL) state. We provide numerical evidence based on exact diagonalization to show that this CSL state is stabilized in an extended region of the parameter space and can be viewed as a lattice version of the Halperin 221 fractional quantum Hall state of two-component bosons. We also construct a parton wave function for this CSL state and demonstrate that its variational energies are in good agreement with exact diagonalization results. The parton description further supports that the CSL is characterized by a chiral edge conformal field theory of the SU(3)1\mathrm{SU}{(3)}_{1}SU(3)1​ Wess-Zumino-Witten type.
We perform an exact-diagonalization study of the spin-12\frac{1}{2}21​ XXZ Heisenberg antiferromagnet on the kagome lattice at finite magnetization m=23m=\frac{2}{3}m=32​ with an emphasis on the XYXYXY point (Jz=0)({J}_{z}=0)(Jz​=0) and in the presence of a small chiral term. Recent analytic work by Kumar et al. [K. Kumar, K. Sun, and E. Fradkin, Phys. Rev. B 90, 174409 (2014)] on the same model, using a newly developed flux attachment transformation, predicts a plateau at this value of the magnetization described by a chiral spin liquid (CSL) with a spin Hall conductance of ${\ensuremath{\sigma}}_{xy}=\frac{1}{2}$. Such a state is topological in nature, has a ground-state degeneracy, and exhibits fractional excitations. We analyze the degeneracy structure in the low-energy manifold, identify the candidate topological states, and use them to compute the modular matrices and Chern numbers, all of which strongly agree with expected theoretical behavior for the ${\ensuremath{\sigma}}_{xy}=\frac{1}{2}$ CSL. In the limit of zero chirality, we find on most (not all) clusters that the topological invariants are still those of a CSL.