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arXiv:1302.4512 · Physical Review B

Possible uniform-flux chiral spin liquid states in the SU(3) ring-exchange model on the triangular lattice

Hsin-Hua Lai

Published May 9, 2013• 4 citations
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Abstract

We consider a SU(3) model with antiferromagnetic three-site ring exchanges, in addition to two-site exchanges, on the triangular lattice. We first present numerical site-factorized state studies on the magnetic ordered states, which show two different three-sublattice-ordered states---the antiferroquadrupolar phase and the standard ${120}^{\ensuremath{\circ}}$ antiferromagnetic phase---along the axis of the strength of the three-site ring exchanges. We further study the model using slave-fermion mean field approaches in which we rewrite the exchange operators in terms of three flavors of fermions. At the mean-field level, we find that the main competing trial states are the trimer state (triangular plaquette state), gapped uniform $\ensuremath{\pi}/3$-flux chiral spin liquid, and gapped uniform $2\ensuremath{\pi}/3$-flux chiral spin liquid. The filled band of the $\ensuremath{\pi}/3$-flux chiral spin liquid has Chern number $+1$, and that of the $2\ensuremath{\pi}/3$-flux chiral spin liquid has Chern number $+2$. We also give the effective Chern-Simons theory for each chiral spin liquid at the mean-field level.

DOI
10.1103/physrevb.87.205111

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We consider a SU(3) model with antiferromagnetic three-site ring exchanges, in addition to two-site exchanges, on the triangular lattice. We first present numerical site-factorized state studies on the magnetic ordered states, which show two different three-sublattice-ordered states---the antiferroquadrupolar phase and the standard 120\ensuremath∘{120}^{\ensuremath{\circ}}120\ensuremath∘ antiferromagnetic phase---along the axis of the strength of the three-site ring exchanges. We further study the model using slave-fermion mean field approaches in which we rewrite the exchange operators in terms of three flavors of fermions. At the mean-field level, we find that the main competing trial states are the trimer state (triangular plaquette state), gapped uniform \ensuremathπ/3\ensuremath{\pi}/3\ensuremathπ/3-flux chiral spin liquid, and gapped uniform 2\ensuremathπ/32\ensuremath{\pi}/32\ensuremathπ/3-flux chiral spin liquid. The filled band of the \ensuremathπ/3\ensuremath{\pi}/3\ensuremathπ/3-flux chiral spin liquid has Chern number +1+1+1, and that of the 2\ensuremathπ/32\ensuremath{\pi}/32\ensuremathπ/3-flux chiral spin liquid has Chern number +2+2+2. We also give the effective Chern-Simons theory for each chiral spin liquid at the mean-field level.
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In this analytical mean-field study we construct and analyze a chiral SU(3) spin model on the bi- partite honeycomb lattice. Assigning the fundamental and antifundamental representations of SU(3) to the two sublattices, as required to form a singlet on every bond, together with the choice N = 3, fixes the on-site degree of freedom to a qutrit and the two-site coupling to the SU(3) Heisenberg exchange. By a standard representation-theoretic argument, the single-site reduced entropy equals ln 3 exactly in any finite-size singlet ground state. We show that the natural make-before-break ring exchange around a hexagonal plaquette must advance the site states by two positions to respect the alternation of conjugate representations, and that its time-reversal-odd part decomposes exactly into SU(3) scalar chiralities on the two sublattice triangles of the hexagon, leaving a one-parameter model. A fermionic parton mean-field analysis, with a single U(1) gauge constraint, places the parton bands at the Heisenberg point in a topologically trivial gapped state and shows, through an explicit Wick decoupling of the six-site ring exchange, that the chiral term generates a competing Haldane flux; enforcing the constraint self-consistently, the gap inverts beyond a finite coupling into a phase in which each of the three color bands carries unit Chern number. Integrating out the partons yields an SU(3)1 chiral spin liquid. The candidate phase carries total quantum dimension√3, three anyon sectors with Z3 fusion, chiral central charge two, and topological entanglement entropy ln √3. We present the exact operator identities as an algebraic foundation, and the SU(3)1 identification as a mean-field prediction, for future high-performance tensor-network investigations, which we identify as the necessary test of the proposed phase.
Motivated by recent experiments on Ba3{}_{3}3​NiSb2{}_{2}2​O9{}_{9}9​, we investigate possible quantum spin liquid ground states for spin S=1S=1S=1 Heisenberg models on the triangular lattice. We use variational Monte Carlo techniques to calculate the energies of microscopic spin liquid wave functions where spin is represented by three flavors of fermionic spinon operators. These energies are compared with the energies of various competing three-sublattice ordered states. Our approach shows that the antiferromagnetic Heisenberg model with biquadratic term and single-ion anisotropy does not have a low-temperature spin liquid phase. However, for an SU(3)-invariant model with sufficiently strong ring-exchange terms, we find a paired chiral quantum spin liquid with a Fermi surface of deconfined spinons that is stable against all types of ordering patterns we considered. We discuss the physics of this exotic spin liquid state in relation to the recent experiment and suggest new ways to test this scenario.
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We report the study of phases and transitions of SU(N) Hofstadter-Hubbard model subject to commensurate magnetic field on the triangular lattice. At filling one fermion per site, for the number of fermion flavors 2 <= N <= 8, we identify three distinct phases and calculate critical interaction strength from parton large-N mean-field approximation. Integer quantum Hall, chiral spin liquid, and valence bond solid states could be realized upon varying the Hubbard interaction U and the number of flavor N . We construct the critical theory for the putative continuous transition from quantum Hall states to chiral spin liquid and calculate the critical transport behavior using quantum Boltzmann equations for general N . These results could be validated in synthetic systems such as moire superlattices and cold atom platforms.