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arXiv:1409.5870 · Physical Review B

Non-symmetry-breaking ground state of the S=1 Heisenberg model on the kagome lattice

Satoshi Nishimoto, Masaaki Nakamura

Published Oct 21, 2015• 23 citations
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Abstract

The $S=1$ antiferromagnetic Heisenberg model on a kagome lattice is studied using the density-matrix renormalization-group method. To identify the ground state, we take four kinds of clusters into account; periodic, cylindrical, and two open ones. The hexagonal-singlet solid (HSS) and triangular valence-bond solid (TVBS) states are artificially generated by modulating edge shapes of the open clusters. We find that the energy per site of the HSS state is ${e}_{0}={E}_{0}/(NJ)=\ensuremath{-}1.410\phantom{\rule{0.16em}{0ex}}95$, which is readily lower than that of the TVBS state (${e}_{0}=\ensuremath{-}1.3912\ifmmode\pm\else\textpm\fi{}0.0025$). This agrees well with those of the cylindrical (${e}_{0}=\ensuremath{-}1.4122$) and periodic (${e}_{0}=\ensuremath{-}1.4111$) clusters where no assumption as to the ordering is posed. We further calculate the dimer-dimer correlation functions as well as the entanglement entropy of cylindrical clusters and conclude that a non-symmetry-breaking state is the ground state of the $S=1$ kagome Heisenberg model. Finally, we estimate the singlet-triplet energy gap: The HSS state has $\mathrm{\ensuremath{\Delta}}/J=0.178\ifmmode\pm\else\textpm\fi{}0.005$, whereas the higher TVBS state has a larger one $\mathrm{\ensuremath{\Delta}}/J=0.280$.

DOI
10.1103/physrevb.92.140412

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The S=1S=1S=1 antiferromagnetic Heisenberg model on a kagome lattice is studied using the density-matrix renormalization-group method. To identify the ground state, we take four kinds of clusters into account; periodic, cylindrical, and two open ones. The hexagonal-singlet solid (HSS) and triangular valence-bond solid (TVBS) states are artificially generated by modulating edge shapes of the open clusters. We find that the energy per site of the HSS state is e0=E0/(NJ)=\ensuremath−1.41095{e}_{0}={E}_{0}/(NJ)=\ensuremath{-}1.410\phantom{\rule{0.16em}{0ex}}95e0​=E0​/(NJ)=\ensuremath−1.41095, which is readily lower than that of the TVBS state (e0=\ensuremath−1.3912\ifmmode±\else\textpm\fi0.0025{e}_{0}=\ensuremath{-}1.3912\ifmmode\pm\else\textpm\fi{}0.0025e0​=\ensuremath−1.3912\ifmmode±\else\textpm\fi0.0025). This agrees well with those of the cylindrical (e0=\ensuremath−1.4122{e}_{0}=\ensuremath{-}1.4122e0​=\ensuremath−1.4122) and periodic (e0=\ensuremath−1.4111{e}_{0}=\ensuremath{-}1.4111e0​=\ensuremath−1.4111) clusters where no assumption as to the ordering is posed. We further calculate the dimer-dimer correlation functions as well as the entanglement entropy of cylindrical clusters and conclude that a non-symmetry-breaking state is the ground state of the S=1S=1S=1 kagome Heisenberg model. Finally, we estimate the singlet-triplet energy gap: The HSS state has \ensuremathΔ/J=0.178\ifmmode±\else\textpm\fi0.005\mathrm{\ensuremath{\Delta}}/J=0.178\ifmmode\pm\else\textpm\fi{}0.005\ensuremathΔ/J=0.178\ifmmode±\else\textpm\fi0.005, whereas the higher TVBS state has a larger one \ensuremathΔ/J=0.280\mathrm{\ensuremath{\Delta}}/J=0.280\ensuremathΔ/J=0.280.
Using series expansions around the dimer limit, we find that the ground state of the spin-1/2 Heisenberg antiferromagnet on the kagome lattice appears to be a valence bond crystal (VBC) with a 36 site unit cell, and ground-state energy per site $E=\ensuremath{-}0.433\ifmmode\pm\else\textpm\fi{}0.001\phantom{\rule{0.3em}{0ex}}\mathrm{J}$. It consists of a honeycomb lattice of ``perfect hexagons.'' The energy difference between the ground state and other ordered states with the maximum number of perfect hexagons, such as a stripe-ordered state, is of order 0.001J0.001\phantom{\rule{0.3em}{0ex}}\mathrm{J}0.001J. The expansion is also done for the 36 site system with periodic boundary conditions; its energy per site is $0.005\ifmmode\pm\else\textpm\fi{}0.001\phantom{\rule{0.3em}{0ex}}\mathrm{J}$ lower than the infinite system, consistent with exact diagonalization results. Every unit cell of the VBC has two singlet states whose degeneracy is not lifted to sixth order in the expansion. We estimate this energy difference to be less than 0.001J0.001\phantom{\rule{0.3em}{0ex}}\mathrm{J}0.001J. The dimerization order parameter is found to be robust. Two leading orders of perturbation theory give lowest triplet excitations to be dispersionless and confined to the perfect hexagons.
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.
Motivated by a recent experiment on Rb2Cu3SnF12{\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}Rb2​Cu3​SnF12​, where spin-1/2 Cu2+{\text{Cu}}^{2+}Cu2+ moments reside on the layers of kagome-like lattices, we investigate quantum ground states of the antiferromagnetic Heisenberg model on a series of deformed kagome lattices. The deformation is characterized by a weaker exchange coupling $\ensuremath{\alpha}J$ on certain lattice links appropriate for Rb2Cu3SnF12{\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}Rb2​Cu3​SnF12​ with $\ensuremath{\alpha}=1$ corresponding to the ideal kagome lattice. In particular, we study possible valence bond solid phases using the perturbation theory around isolated dimer limits, dimer series expansion, and self-consistent bond operator mean-field theory. It is shown that the valence bond solid phase with a 36-site unit cell of the ideal kagome lattice is quite sensitive to a small lattice distortion as the kind discovered in Rb2Cu3SnF12{\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}Rb2​Cu3​SnF12​. As a result, we find that a more likely quantum ground state in Rb2Cu3SnF12{\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}Rb2​Cu3​SnF12​ is the valence bond solid phase with a 12-site unit cell, where six dimers form a pinwheel structure, leading to strong modification of the elementary triplet and singlet excitation spectra in the deformed kagome lattices.
We study the ground-state phase diagram of the spin-1/21/21/2 antiferromagnetic Heisenberg model on the square-kagome lattice using infinite projected entangled-pair states (iPEPS). By systematically varying the ratio of exchange couplings on triangular and square plaquettes, we establish a complete quantum phase diagram in the thermodynamic limit. In the intermediate-coupling regime, we identify four distinct nonmagnetic phases that are unambiguously characterized as valence-bond crystals (VBCs) by their symmetry-inequivalent patterns of strong and weak spin-spin correlations. These include a plaquette crossed-dimer VBC, a loop-six VBC stabilized around the isotropic point, a generalized pinwheel VBC with reduced rotational symmetry, and a decorated loop-six VBC proximate to ferrimagnetic order. We determine the phase boundaries using a combination of bond-resolved correlation functions, entanglement entropy, and magnetization. For transitions not accompanied by sharp entanglement signatures, we extract the spin gap from finite-field simulations, allowing us to distinguish gapped and gapless VBC phases. At larger coupling ratios, the system undergoes transitions into imperfect and perfect ferrimagnetic states. Our results resolve long-standing ambiguities in the square-kagome Heisenberg model and provide a quantitatively reliable reference for future theoretical and experimental studies of frustrated quantum magnets.
We perform a density-matrix renormalization group (DMRG) study of the S=1/2 Heisenberg antiferromagnet on the kagome lattice to identify the conjectured spin liquid ground state. Exploiting SU(2) spin symmetry, which allows us to keep up to 16,000 DMRG states, we consider cylinders with circumferences up to 17 lattice spacings and find a spin liquid ground state with an estimated per site energy of -0.4386(5), a spin gap of 0.13(1), very short-range decay in spin, dimer and chiral correlation functions and finite topological entanglement gamma consistent with gamma=ld 2, ruling out gapless, chiral or non-topological spin liquids. At the same time, DMRG results provide strong evidence for a gapped topological Z2 spin liquid.
Entanglement renormalization techniques are applied to numerically investigate the ground state of the spin-1/2 Heisenberg model on a kagome lattice. Lattices of N={36,144, infinity} sites with periodic boundary conditions are considered. For the infinite lattice, the best approximation to the ground state is found to be a valence bond crystal with a 36-site unit cell, compatible with a previous proposal. Its energy per site, E=-0.432 21, is an exact upper bound and is lower than the energy of any previous (gapped or algebraic) spin liquid candidate for the ground state.