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arXiv:1601.00959 · Physical review. B./Physical review. B

Plaquette order in the SU(6) Heisenberg model on the honeycomb lattice

Pierre Nataf, Miklós Lajkó, Philippe Corboz, Andreas Martin Läuchli, Karlo Penc, Frédéric Mila

Published May 26, 2016• 39 citations
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Abstract

We revisit the SU(6) Heisenberg model on the honeycomb lattice, which has been predicted to be a chiral spin liquid by mean-field theory [G. Szirmai et al., Phys. Rev. A 84, 011611(R) (2011)]. Using exact diagonalizations of finite clusters, infinite projected entangled pair state simulations, and variational Monte Carlo simulations based on Gutzwiller projected wave functions, we provide strong evidence that the model with one particle per site and nearest-neighbor exchange actually develops plaquette order. This is further confirmed by the investigation of the model with a ring-exchange term, which shows that there is a transition between the plaquette state and the chiral state at a finite value of the ring-exchange term.

DOI
10.1103/physrevb.93.201113

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We revisit the SU(6) Heisenberg model on the honeycomb lattice, which has been predicted to be a chiral spin liquid by mean-field theory [G. Szirmai et al., Phys. Rev. A 84, 011611(R) (2011)]. Using exact diagonalizations of finite clusters, infinite projected entangled pair state simulations, and variational Monte Carlo simulations based on Gutzwiller projected wave functions, we provide strong evidence that the model with one particle per site and nearest-neighbor exchange actually develops plaquette order. This is further confirmed by the investigation of the model with a ring-exchange term, which shows that there is a transition between the plaquette state and the chiral state at a finite value of the ring-exchange term.
We show that, when NNN is a multiple of 6 (N=6mN=6mN=6m, where mmm is an integer), the SU(N)\mathrm{SU}\phantom{\rule{0.16em}{0ex}}{\left(N\right)}_{}SU(N)​ Heisenberg model on the honeycomb lattice with mmm particles per site has a clear tendency toward chiral order as soon as $m\ensuremath{\ge}2$. This conclusion has been reached by a systematic variational Monte Carlo investigation of Gutzwiller projected wave functions as a function of mmm between the case of one particle per site (m=1m=1m=1), for which the ground state has recently been shown to be in a plaquette singlet state, and the $m\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, where a mean-field approach has established that the ground state has chiral order. This demonstrates that the chiral phase can indeed be stabilized for not too large values of mmm, opening the way to its experimental realizations in other lattices.
The SU(4) Heisenberg model can serve as a low-energy model of the Mott insulating state in materials where the spins and orbitals are highly symmetric or in systems of alkaline-earth atoms on an optical lattice. Recently, it has been argued that on the honeycomb lattice, the model exhibits a unique spin-orbital liquid phase with an algebraic decay of correlations [P. Corboz et al., Phys. Rev. X 2, 041013 (2012)]. Here we study the instability of the algebraic spin-orbital liquid toward the spontaneous formation of SU(4) singlet plaquettes (tetramerization). Using a variational Monte Carlo approach to evaluate the projected wave function of fermions with $\ensuremath{\pi}$-flux state, we find that the algebraic liquid is robust and that a small finite value of the next-nearest exchange is needed to induce tetramerization. We also studied the phase diagram of a model which interpolates between the nearest-neighbor Heisenberg model and a Hamiltonian for which the singlet-plaquette-product state is an exact ground state.
Conflicting predictions have been made for the ground state of the SU(3) Heisenberg model on the honeycomb lattice: Tensor network simulations found a plaquette order [Zhao et al., Phys. Rev. B 85, 134416 (2012)], where singlets are formed on hexagons, while linear flavor-wave theory suggested a dimerized, color-ordered state [Lee and Yang, Phys. Rev. B 85, 100402 (2012)]. In this work we show that the former state is the true ground state by a systematic study with infinite projected-entangled pair states (iPEPS), for which the accuracy can be systematically controlled by the so-called bond dimension DDD. Both competing states can be reproduced with iPEPS by using different unit cell sizes. For small DDD the dimer state has a lower variational energy than the plaquette state; however, for large DDD it is the latter which becomes energetically favorable. The plaquette formation is also confirmed by exact diagonalizations and variational Monte Carlo studies, according to which both the dimerized and plaquette states are nonchiral flux states.
We present a comprehensive computational study of the phase diagram of the frustrated S=1/2S=1/2S=1/2 Heisenberg antiferromagnet on the honeycomb lattice, with second-nearest (J2)({J}_{2})(J2​) and third-neighbor (J3)({J}_{3})(J3​) couplings. Using a combination of exact diagonalizations (EDs) of the original spin model, of the Hamiltonian projected into the nearest-neighbor short-range valence-bond basis, and of an effective quantum dimer model, as well as a self-consistent cluster mean-field theory, we determine the boundaries of several magnetically ordered phases in the region ${J}_{2},{J}_{3}\ensuremath{\in}[0,1]$, and find a sizable magnetically disordered region in between. We characterize part of this magnetically disordered phase as a plaquette valence-bond crystal phase. At larger J2{J}_{2}J2​, we locate a sizable region in which staggered valence-bond crystal correlations are found to be important, either due to genuine valence-bond crystal (VBC) ordering or as a consequence of magnetically ordered phases, which break lattice rotational symmetry. Furthermore, we find that a particular parameter-free Gutzwiller projected tight-binding wave function has remarkably accurate energies compared to finite-size extrapolated ED energies along the transition line from conventional N\'eel to plaquette VBC phases, a fact that points to possibly interesting critical behavior---such as a deconfined critical point---across this transition. We also comment on the relevance of this spin model to model the spin liquid region found in the half filled Hubbard model on the honeycomb lattice.
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.
We study an S = 1/2 Heisenberg model on the honeycomb lattice with first and second neighbor antiferromagnetic exchange (J(1)-J(2) model), employing exact diagonalization in both the S(z) = 0 basis and nearest neighbor singlet valence bond (NNVB) basis. We find that for 0.2 < J(2)/J(1) < 0.3, the NNVB basis gives a proper description of the ground state in comparison with the exact results. By analyzing the dimer-dimer as well as the plaquette-plaquette correlations and also defining appropriate structure factors, we investigate possible symmetry breaking states as the candidates for the ground state in the frustrated region. We provide a body of evidence in favor of plaquette valence bond ordering for 0.2 < J(2)/J(1) < 0.3. By further increasing the ratio J(2)/J(1), this state undergoes a transition to the staggered dimerized state.