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arXiv:1204.6682 · Physical Review B

Simplex solids in SU( N ) Heisenberg models on the kagome and checkerboard lattices

Philippe Corboz, Karlo Penc, Frédéric Mila, Andreas Martin Läuchli

Published Jul 23, 2012• 70 citations
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Abstract

We present a numerical study of the SU($N$) Heisenberg model with the fundamental representation at each site for the kagome lattice (for $N=3$) and the checkerboard lattice (for $N=4$), which are the line graphs of the honeycomb and square lattices and thus belong to the class of bisimplex lattices. Using infinite projected entangled-pair states and exact diagonalizations, we show that in both cases the ground state is a simplex solid state with a twofold degeneracy, in which the $N$ spins belonging to a simplex (i.e., a complete graph) form a singlet. Theses states can be seen as generalizations of valence bond solid states known to be stabilized in certain SU(2) spin models.

DOI
10.1103/physrevb.86.041106

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We present a numerical study of the SU(NNN) Heisenberg model with the fundamental representation at each site for the kagome lattice (for N=3N=3N=3) and the checkerboard lattice (for N=4N=4N=4), which are the line graphs of the honeycomb and square lattices and thus belong to the class of bisimplex lattices. Using infinite projected entangled-pair states and exact diagonalizations, we show that in both cases the ground state is a simplex solid state with a twofold degeneracy, in which the NNN spins belonging to a simplex (i.e., a complete graph) form a singlet. Theses states can be seen as generalizations of valence bond solid states known to be stabilized in certain SU(2) spin models.
We study the structure of quantum ground states of simplex solid models, which are generalizations of the valence bond construction for quantum antiferromagnets originally proposed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) [Phys. Rev. Lett. 59, 799 (1987)]. Whereas the AKLT states are created by application of bond singlet operators for SU(2) spins, the simplex solid construction is based on N-simplex singlet operators for SU(N) spins. In both cases, a discrete one-parameter family of translationally-invariant models with exactly solvable ground states is defined on any regular lattice, and the equal time ground state correlations are given by the finite temperature correlations of an associated classical model on the same lattice, owing to the product form of the wave functions when expressed in a CP^{N-1} coherent state representation. We study these classical companion models via a mix of Monte Carlo simulations, mean-field arguments, and low-temperature effective field theories. Our analysis reveals that the ground states of SU(4) edge- and SU(8) face-sharing cubic lattice simplex solid models are long range ordered for sufficiently large values of the discrete parameter, whereas the ground states of the SU(3) models on the kagome (2D) and hyperkagome (3D) lattices are always quantum disordered. The kagome simplex solid exhibits strong local order absent in its three-dimensional hyperkagome counterpart, a contrast that we rationalize with arguments similar to those leading to `order by disorder'.
I define a set of wave functions for SU(N)\mathrm{SU}(N)SU(N) lattice antiferromagnets, analogous to the valence bond solid states of Affleck et al. [Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477 (1988)], in which the singlets are extended over NNN-site simplixes. As with the valence bond solids, the new simplex solid (SS) states are extinguished by certain local projection operators, allowing one to construct Hamiltonians with local interactions which render the SS states exact ground states. Using a coherent state representation, I show that the quantum correlations in each SS state are calculable as the finite temperature correlations of an associated classical model with NNN-spin interactions on the same lattice. In three and higher dimensions, the SS states can spontaneously break SU(N)\mathrm{SU}(N)SU(N) and exhibit NNN-sublattice long-ranged order as a function of a discrete parameter which fixes the local representation of SU(N)\mathrm{SU}(N)SU(N). I analyze this transition using a classical mean field approach. For N>2N>2N>2, the ordered state is selected via an ``order by disorder'' mechanism. As in the Affleck-Kennedy-Lieb-Tasaki case, the bulk representations fractionalize at an edge, and the ground state entropy is proportional to the volume of the boundary.
Abstract. We study the structure of quantum ground states of simplex solid models, which are generalizations of the valence bond construction for quantum antiferromagnets originally proposed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) (1987 Phys. Rev. Lett . 59 799). Whereas the AKLT states are created by application of bond singlet operators for spins, the simplex solid construction is based on N -simplex singlet operators for spins. In both cases, a discrete one-parameter family of translationally-invariant models with exactly solvable ground states is defined on any regular lattice, and the equal time ground state correlations are given by the finite temperature correlations of an associated classical model on the same lattice, owing to the product form of the wave functions when expressed in a coherent state representation. We study these classical companion models via a mix of Monte Carlo simulations, mean-field arguments, and low-temperature effective field theories. Our analysis reveals that the ground states of edge- and face-sharing cubic lattice simplex solid models are long range ordered for sufficiently large values of the discrete parameter, whereas the ground states of the models on the kagome (2D) and hyperkagome (3D) lattices are always quantum disordered. The kagome simplex solid exhibits strong local order absent in its three-dimensional hyperkagome counterpart, a contrast that we rationalize with arguments similar to those leading to ‘order by disorder’.
We investigate the spin-1/21/21/2 Heisenberg antiferromagnet on the ruby lattice with uniform first- and second-neighbor interactions, which forms a two-dimensional network of corner-sharing tetrahedra. Using infinite projected entangled pair states (iPEPS), we study the ground state of the system to find that it assumes a gapped threefold-degenerate simplex crystal ground state, with strong singlets formed on pairs of neighboring triangles. We argue that the formation of the simplex singlet ground state at the isotropic point relates to the weak inter-triangle coupling limit where an effective spin-chirality Hamiltonian on the honeycomb lattice exhibits an extensively degenerate ground state manifold of singlet coverings at the mean-field level. Under an applied Zeeman field, the iPEPS simulations uncover magnetization plateaus at m/ms=0,1/3,1/2,m/m_s = 0, 1/3, 1/2,m/ms​=0,1/3,1/2, and 2/32/32/3, separated by intermediate supersolid phases, all breaking the sixfold rotational symmetry of the lattice. Unlike the checkerboard lattice, these plateaus cannot be described by strongly localized magnons.
We study numerically the spin-1/2 XXZ model in a field on an infinite kagome lattice. We use different algorithms based on infinite projected entangled pair states (iPEPSs) for this, namely, (i) an approach with simplex tensors and a 9-site unit cell, and (ii) an approach based on coarse-graining three spins in the kagome lattice and mapping it to a square-lattice model with local and nearest-neighbor interactions, with the usual PEPS tensors, 6- and 12-site unit cells. Similarly to our previous calculation at the SU(2)-symmetric point (Heisenberg Hamiltonian), for any anisotropy from the Ising limit to the XY limit, we also observe the emergence of magnetization plateaus as a function of the magnetic field, at mz=13{m}_{z}=\frac{1}{3}mz​=31​ using 6-, 9-, and 12-site PEPS unit cells, and at mz=19,59{m}_{z}=\frac{1}{9},\phantom{\rule{0.28em}{0ex}}\frac{5}{9}mz​=91​,95​, and 79\frac{7}{9}97​ using a 9-site PEPS unit cell, the latter setup being able to accommodate $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ solid order. We also find that, at mz=13{m}_{z}=\frac{1}{3}mz​=31​, (lattice) nematic and $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC-order states are degenerate within the accuracy of the nine-site simplex method, for all anisotropy. The 6- and 12-site coarse-grained PEPS methods produce almost-degenerate nematic and $1\ifmmode\times\else\texttimes\fi{}2$ VBC-solid orders. We also find that, within our accuracy, the six-site coarse-grained PEPS method gives slightly lower energies, which can be explained by the larger amount of entanglement this approach can handle, even in cases where the PEPS unit cell is not commensurate with the expected ground-state unit cell. Furthermore, we do not observe chiral spin liquid behaviors at and close to the XY point, as has been recently proposed. Our results are the first tensor network investigations of the XXZ model in a field and reveal the subtle competition between nearby magnetic orders in numerical simulations of frustrated quantum antiferromagnets, as well as the delicate interplay between energy optimization and symmetry in tensor network numerical simulations.
We investigate the ground-state properties of a spin-1 kagome antiferromagnetic Heisenberg model using tensor-network (TN) methods. We obtain the energy per site ${e}_{0}=\ensuremath{-}1.410\phantom{\rule{0.16em}{0ex}}90(2)$, with D∗=8{D}^{*}=8D∗=8 multiplets retained (i.e., a bond dimension of D=24)D=24)D=24), and ${e}_{0}=\ensuremath{-}1.4116(4)$ from large-DDD extrapolation, by accurate TN calculations directly in the thermodynamic limit. The symmetry between the two kinds of triangles is spontaneously broken, with a relative energy difference of $\ensuremath{\delta}\ensuremath{\approx}19%$, i.e, there is a trimerization (simplex) valence-bond order in the ground state. The spin-spin, dimer-dimer, and chirality-chirality correlation functions are found to decay exponentially with a rather short correlation length, showing that the ground state is gapped. We thus identify the ground state to be a simplex valence-bond crystal. We also discuss the spin-1 bilinear-biquadratic Heisenberg model on a kagome lattice, and determine its ground-state phase diagram. Moreover, we implement non-Abelian symmetries, here spin SU(2), in the TN algorithm, which improves the efficiency greatly and provides insight into the tensor structures.