Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9901068 · Physical Review Letters

Long-Range Néel Order in the Triangular Heisenberg Model

Luca Capriotti, A. E. Trumper, Sandro Sorella

Published May 10, 1999• 463 citations
arXivHTMLPDF

Abstract

We have studied the Heisenberg model on the triangular lattice using quantum Monte Carlo techniques (up to 144 sites) and exact diagonalization (up to 36 sites). By studying the spin gap as a function of the system size we have obtained robust evidence for a gapless spectrum, confirming the existence of long-range N\'eel order. Our best estimate is that in the thermodynamic limit the order parameter ${m}^{\ifmmode\dagger\else\textdagger\fi{}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.41\ifmmode\pm\else\textpm\fi{}0.02$ is reduced by about $59%$ from its classical value and the ground state energy per site is ${e}_{0}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}0.5458\ifmmode\pm\else\textpm\fi{}0.0001$ in units of the exchange coupling. We have identified the ground state correlations that are important at short distances.

DOI
10.1103/physrevlett.82.3899

Related papers

Ground state properties of the S=1/2 Heisenberg antiferromagnet on the triangular lattice

Luca Capriotti, A. E. Trumper, Sandro Sorella

We investigate the ground state properties of the S=1/2 Heisenberg antiferromagnet on the triangular lattice using mainly exact diagonalization (for lattices up to N=36 sites) and the recently developed Green Function Monte Carlo with stochastic reconfiguration (up to N=144). A systematic size scaling of the order parameter and of the spin gap gives strong evidences for the presence of long-range Néel order in the ground state of this highly frustrated model.

Journal of Applied Physics•May 1, 2000•2 citations

From the triangular to the kagome lattice: Following the footprints of the ordered state

Liliana Arrachea, Luca Capriotti, Sandro Sorella

We study the spin-$1∕2$ Heisenberg model in a lattice that interpolates between the triangular and the kagome lattices. The exchange interaction along the bonds of the kagome lattice is $J$, and the one along the bonds connecting kagome and nonkagome sites is ${J}^{\ensuremath{'}}$, so that ${J}^{\ensuremath{'}}=J$ corresponds to the triangular limit and ${J}^{\ensuremath{'}}=0$ to the kagome one. We use variational and exact diagonalization techniques. We analyze the behavior of the order parameter for the antiferromagnetic phase of the triangular lattice, the spin gap, and the structure of the spin excitations as functions of ${J}^{\ensuremath{'}}∕J$. Our results indicate that the antiferromagnetic order is not affected by the reduction of ${J}^{\ensuremath{'}}$ down to ${J}^{\ensuremath{'}}∕J\ensuremath{\simeq}0.2$. Below this value, antiferromagnetic correlations grow weaker, a description of the ground state in terms of a N\'eel phase renormalized by quantum fluctuations becomes inadequate, and the finite-size spectra develop features that are not compatible with antiferromagnetic ordering. However, this phase does not appear to be connected to the kagome phase as well, as the low-energy spectra do not evolve with continuity for ${J}^{\ensuremath{'}}\ensuremath{\rightarrow}0$ to the kagome limit. In particular, for any nonzero value of ${J}^{\ensuremath{'}}$, the latter interaction sets the energy scale for the low-lying spin excitations, and a gapless triplet spectrum, destabilizing the kagome phase, is expected.

arXiv:cond-mat/0311538•Physical Review B•Jun 29, 2004•10 citations

Ground-state properties of the spin-\frac{1}{2} antiferromagnetic Heisenberg model on the triangular lattice: a variational study based on entangled-plaquette states

Fabio Mezzacapo, J. I. Cirac

We study, on the basis of the general entangled-plaquette variational ansatz, the ground-state (GS) properties of the spin- antiferromagnetic Heisenberg model on the triangular lattice. Our numerical estimates are in good agreement with available exact results and comparable, for large system sizes, to those computed via the best alternative numerical approaches, or by means of variational schemes based on specific (i.e. incorporating problem-dependent terms) trial wave functions (WFs). The extrapolation to the thermodynamic limit of our results for lattices comprising up to N =324 spins yields an upper bound of the GS energy per site (in units of the exchange coupling) of − 0.5458(2) [−0.4074(1) for the XX model], while the estimated infinite-lattice order parameter is 0.3178(5) (i.e. ∼64% of the classical value).

arXiv:1006.4480•New Journal of Physics•Oct 26, 2010•33 citations

Spin-liquid phase of the multiple-spin exchange Hamiltonian on the triangular lattice

Grégoire Misguich, C. Lhuillier, B. Bernu +1 more

We performed an exact diagonalization study of the spin-liquid phase of the multiple-spin exchange model on a triangular lattice. It is characterized by no N\'eel long-range order, short-ranged magnetic correlations, and a spin gap. We found no long-range order in any local order parameter we investigated (chiral, dimer, etc.) The probable asymptotic ground-state degeneracy is discussed. We argue that it could be of topological origin and that the system is probably not a chiral spin liquid. A possible relation to the Affleck-Kennedy-Lieb-Tasaki, or valence-bond solid phase is discussed.

arXiv:cond-mat/9812329•Physical review. B, Condensed matter•Jul 1, 1999•201 citations

Magnetization of the spin- 12 Heisenberg antiferromagnet on the triangular lattice

Qian Li, Li Hong, Jize Zhao +2 more

After decades of debate, now there is a rough consensus that at zero temperature the spin-$1/2$ Heisenberg antiferromagnet on the triangular lattice is three-sublattice ${120}^{\ensuremath{\circ}}$ magnetically ordered, in contrast to a quantum spin liquid as originally proposed. However, there remains considerable discrepancy in the magnetization reported among various methods. To resolve this issue, in this work we revisit this model by the tensor-network state algorithm. The ground-state energy per bond ${E}_{b}$ and magnetization per spin ${M}_{0}$ in the thermodynamic limit are obtained with high precision. The former is estimated to be ${E}_{b}=\ensuremath{-}0.18334(10)$. This value agrees well with that from the series expansion. The three-sublattice magnetic order is firmly confirmed and the magnetization is determined as ${M}_{0}=0.161(5)$. It is about $32%$ of its classical value and slightly below the lower bound from the series expansion. In comparison with the best estimated value by Monte Carlo and density-matrix renormalization group, our result is about $20%$ smaller. This magnetic order is consistent with further analysis of the three-body correlation. Our work thus provides benchmark results for this prototypical model.

arXiv:2009.03765•Physical review. B./Physical review. B•May 16, 2022•23 citations

Spectral Mapping Theorem between SU(N) and SU(Ns/N) Heisenberg Models and Exact Diagonalization of SU(4) and SU(7) Long-Range Ordered Phases on the Triangular Lattice

Samuel Gozel, Pierre Nataf

We first show that, for the SU(N) Heisenberg model with isotropic two-site interactions on an arbitrary lattice of Ns sites, the full spectrum in its singlet sector (when N divides Ns) is the negative of the full spectrum in its singlet sector of the SU(Ns/N) Heisenberg model defined on the same lattice. This theorem, valid for models with the fundamental irreducible representation (irrep) at each lattice site, follows naturally from a transposition operation in the basis of Standard Young Tableaux (SYTs). Second, this basis -- previously recognized as particularly convenient for exact diagonalization (ED) of SU(N) Heisenberg models -- admits an efficient, memory-saving implementation, thereby enabling large-scale diagonalizations for unprecedented system sizes Ns and numbers of colors N>3. We apply these ideas to an ED study of the SU(4) and SU(7) Heisenberg models on the triangular lattice (TL), extending the calculations up to Ns=28 sites. In particular, starting from the SU(4) case, whose nearest-neighbor ground state remains debated, we confirm that next-nearest-neighbor couplings stabilize four-sublattice long-range order. We further find that increasing the number of colors from N=4 to N=7 provides another route toward a Néel-like ground state, even without such additional couplings. N = 4 to N = 7 provides another route toward a Néel-like ground state, even without such additional couplings.

arXiv:2610.15162•Oct 10, 2026•0 citations
We have studied the Heisenberg model on the triangular lattice using quantum Monte Carlo techniques (up to 144 sites) and exact diagonalization (up to 36 sites). By studying the spin gap as a function of the system size we have obtained robust evidence for a gapless spectrum, confirming the existence of long-range N'eel order. Our best estimate is that in the thermodynamic limit the order parameter m\ifmmode†\else\textdagger\fi=0.41\ifmmode±\else\textpm\fi0.02{m}^{\ifmmode\dagger\else\textdagger\fi{}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.41\ifmmode\pm\else\textpm\fi{}0.02m\ifmmode†\else\textdagger\fi=0.41\ifmmode±\else\textpm\fi0.02 is reduced by about 5959%59 from its classical value and the ground state energy per site is e0=\ensuremath−0.5458\ifmmode±\else\textpm\fi0.0001{e}_{0}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}0.5458\ifmmode\pm\else\textpm\fi{}0.0001e0​=\ensuremath−0.5458\ifmmode±\else\textpm\fi0.0001 in units of the exchange coupling. We have identified the ground state correlations that are important at short distances.
We study the spin-1∕21∕21∕2 Heisenberg model in a lattice that interpolates between the triangular and the kagome lattices. The exchange interaction along the bonds of the kagome lattice is JJJ, and the one along the bonds connecting kagome and nonkagome sites is ${J}^{\ensuremath{'}}$, so that ${J}^{\ensuremath{'}}=J$ corresponds to the triangular limit and ${J}^{\ensuremath{'}}=0$ to the kagome one. We use variational and exact diagonalization techniques. We analyze the behavior of the order parameter for the antiferromagnetic phase of the triangular lattice, the spin gap, and the structure of the spin excitations as functions of ${J}^{\ensuremath{'}}∕J$. Our results indicate that the antiferromagnetic order is not affected by the reduction of ${J}^{\ensuremath{'}}$ down to ${J}^{\ensuremath{'}}∕J\ensuremath{\simeq}0.2$. Below this value, antiferromagnetic correlations grow weaker, a description of the ground state in terms of a N\'eel phase renormalized by quantum fluctuations becomes inadequate, and the finite-size spectra develop features that are not compatible with antiferromagnetic ordering. However, this phase does not appear to be connected to the kagome phase as well, as the low-energy spectra do not evolve with continuity for ${J}^{\ensuremath{'}}\ensuremath{\rightarrow}0$ to the kagome limit. In particular, for any nonzero value of ${J}^{\ensuremath{'}}$, the latter interaction sets the energy scale for the low-lying spin excitations, and a gapless triplet spectrum, destabilizing the kagome phase, is expected.
We study, on the basis of the general entangled-plaquette variational ansatz, the ground-state (GS) properties of the spin- antiferromagnetic Heisenberg model on the triangular lattice. Our numerical estimates are in good agreement with available exact results and comparable, for large system sizes, to those computed via the best alternative numerical approaches, or by means of variational schemes based on specific (i.e. incorporating problem-dependent terms) trial wave functions (WFs). The extrapolation to the thermodynamic limit of our results for lattices comprising up to N =324 spins yields an upper bound of the GS energy per site (in units of the exchange coupling) of − 0.5458(2) [−0.4074(1) for the XX model], while the estimated infinite-lattice order parameter is 0.3178(5) (i.e. ∼64% of the classical value).
We performed an exact diagonalization study of the spin-liquid phase of the multiple-spin exchange model on a triangular lattice. It is characterized by no N\'eel long-range order, short-ranged magnetic correlations, and a spin gap. We found no long-range order in any local order parameter we investigated (chiral, dimer, etc.) The probable asymptotic ground-state degeneracy is discussed. We argue that it could be of topological origin and that the system is probably not a chiral spin liquid. A possible relation to the Affleck-Kennedy-Lieb-Tasaki, or valence-bond solid phase is discussed.
After decades of debate, now there is a rough consensus that at zero temperature the spin-1/21/21/2 Heisenberg antiferromagnet on the triangular lattice is three-sublattice ${120}^{\ensuremath{\circ}}$ magnetically ordered, in contrast to a quantum spin liquid as originally proposed. However, there remains considerable discrepancy in the magnetization reported among various methods. To resolve this issue, in this work we revisit this model by the tensor-network state algorithm. The ground-state energy per bond Eb{E}_{b}Eb​ and magnetization per spin M0{M}_{0}M0​ in the thermodynamic limit are obtained with high precision. The former is estimated to be ${E}_{b}=\ensuremath{-}0.18334(10)$. This value agrees well with that from the series expansion. The three-sublattice magnetic order is firmly confirmed and the magnetization is determined as M0=0.161(5){M}_{0}=0.161(5)M0​=0.161(5). It is about 3232%32 of its classical value and slightly below the lower bound from the series expansion. In comparison with the best estimated value by Monte Carlo and density-matrix renormalization group, our result is about 2020%20 smaller. This magnetic order is consistent with further analysis of the three-body correlation. Our work thus provides benchmark results for this prototypical model.
We first show that, for the SU(N) Heisenberg model with isotropic two-site interactions on an arbitrary lattice of Ns sites, the full spectrum in its singlet sector (when N divides Ns) is the negative of the full spectrum in its singlet sector of the SU(Ns/N) Heisenberg model defined on the same lattice. This theorem, valid for models with the fundamental irreducible representation (irrep) at each lattice site, follows naturally from a transposition operation in the basis of Standard Young Tableaux (SYTs). Second, this basis -- previously recognized as particularly convenient for exact diagonalization (ED) of SU(N) Heisenberg models -- admits an efficient, memory-saving implementation, thereby enabling large-scale diagonalizations for unprecedented system sizes Ns and numbers of colors N>3. We apply these ideas to an ED study of the SU(4) and SU(7) Heisenberg models on the triangular lattice (TL), extending the calculations up to Ns=28 sites. In particular, starting from the SU(4) case, whose nearest-neighbor ground state remains debated, we confirm that next-nearest-neighbor couplings stabilize four-sublattice long-range order. We further find that increasing the number of colors from N=4 to N=7 provides another route toward a Néel-like ground state, even without such additional couplings. N = 4 to N = 7 provides another route toward a Néel-like ground state, even without such additional couplings.