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/0311538 · Physical Review B

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

Liliana Arrachea, Luca Capriotti, Sandro Sorella

Published Jun 29, 2004• 10 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.69.224414

Related papers

Long-Range Néel Order in the Triangular Heisenberg Model

Luca Capriotti, A. E. Trumper, Sandro Sorella

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.

arXiv:cond-mat/9901068•Physical Review Letters•May 10, 1999•463 citations

Plaquette-triplon analysis of magnetic disorder and order in a trimerized spin-1 kagome Heisenberg antiferromagnet

Pratyay Ghosh, Akhilesh Kumar Verma, Brijesh Kumar

A spin-1 Heisenberg model on trimerized kagome lattice is studied by doing a low-energy bosonic theory in terms of plaquette triplons defined on its triangular unit cells. The model considered has an intratriangle antiferromagnetic exchange interaction $J$ (set to 1) and two intertriangle couplings ${J}^{\ensuremath{'}}>0$ (nearest neighbor) and ${J}^{\ensuremath{''}}$ (next nearest neighbor; of both signs). The triplon analysis performed on this model investigates the stability of the trimerized singlet ground state (which is exact in the absence of intertriangle couplings) in the ${J}^{\ensuremath{'}}\ensuremath{-}{J}^{\ensuremath{''}}$ plane. It gives a quantum phase diagram that has two gapless antiferromagnetically ordered phases separated by the spin-gapped trimerized singlet phase. The trimerized singlet ground state is found to be stable on ${J}^{\ensuremath{''}}=0$ line (the nearest-neighbor case), and on both sides of it for ${J}^{\ensuremath{''}}\ensuremath{\ne}0$, in an extended region bounded by the critical lines of transition to the gapless antiferromagnetic phases. The gapless phase in the negative ${J}^{\ensuremath{''}}$ region has a coplanar ${120}^{\ensuremath{\circ}}$ antiferromagnetic order with $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ structure. In this phase, all the magnetic moments are of equal length, and the angle between any two of them on a triangle is exactly ${120}^{\ensuremath{\circ}}$. The magnetic lattice in this case has a unit cell consisting of three triangles. The other gapless phase, in the positive ${J}^{\ensuremath{''}}$ region, is found to exhibit a different coplanar antiferromagnetic order with ordering wave vector $\mathbf{q}=(0,0)$. Here, two magnetic moments in a triangle are of the same magnitude, but shorter than the third. While the angle between two short moments is ${120}^{\ensuremath{\circ}}\ensuremath{-}2\ensuremath{\delta}$, it is ${120}^{\ensuremath{\circ}}+\ensuremath{\delta}$ between a short and the long one. Only when ${J}^{\ensuremath{''}}={J}^{\ensuremath{'}}$, their magnitudes become equal and the relative angles ${120}^{\ensuremath{\circ}}$. The magnetic lattice in this $\mathbf{q}=(0,0)$ phase has the translational symmetry of the kagome lattice with triangular unit cells of reduced (isosceles) symmetry. This reduction in the point-group symmetry is found to show up as a difference in the intensities of certain Bragg peaks, whose ratio ${I}_{(1,0)}/{I}_{(0,1)}=4{sin}^{2}(\frac{\ensuremath{\pi}}{6}+\ensuremath{\delta})$ presents an experimental measure of the deviation $\ensuremath{\delta}$ from the ${120}^{\ensuremath{\circ}}$ order.

arXiv:1507.08038•Physical review. B./Physical review. B•Jan 19, 2016•16 citations

Néel order versus spin liquid in quantum Heisenberg antiferromagnets on triangular and Kagomé lattices

B. Bernu, P. Lecheminant, C. Lhuillier +1 more

We have realized a thorough quantum mechanical analysis of an extensive number of low lying levels of the Heisenberg Hamiltonian on small periodic samples. Striking qualitative differences appear between the spectra of the triangular lattice and those of the Kagomé lattice. Symmetries and finite size scaling of the low lying levels of the triangular antiferromagnetic point to an à la Néel symmetry breaking, and the dynamical structure factors corroborate this conclusion. The Kagomé antiferromagnet spectra are qualitatively different: no signature of a symmetry breaking has been exhibited and the dynamical structure factors of the low lying levels display broad bands. The Heisenberg antiferromagnet on a Kagomé lattice is a serious candidate for a spin liquid.

Physica Scripta•Jan 1, 1993•19 citations

Ground states and excitations of a one-dimensionalkagomé-like antiferromagnet

Ch. Waldtmann, H. Kreutzmann, Ulrich Schollwöck +2 more

We study a Heisenberg antiferromagnet (AF) on a one-dimensional strip composed of two rows of corner sharing triangular plaquettes. The geometry of this ladder-like lattice naturally allows for two different exchange couplings, one between pairs of spins on the outer legs (J) and a different one between the spins on the axis and their nearest neighbors on the legs ${(J}^{\ensuremath{'}}).$ For ${J/J}^{\ensuremath{'}}\ensuremath{\lesssim}0.5$ the model is a ferrimagnet. Our main interest is in the region ${J/J}^{\ensuremath{'}}\ensuremath{\gtrsim}0.5,$ where the classical ground state of this system shows the same macroscopic degeneracy as the classical ground state of the antiferromagnet on the kagom\'e lattice. To explore to which extent this similarity between the classical ground states of these two models carries over to their quantum states we have applied exact diagonalization techniques and density-matrix renormalization group (DMRG) methods to our model. Exact diagonalization is restricted to system sizes of up to $N=30$ sites. For ${J/J}^{\ensuremath{'}}=1,$ the results obtained by this technique, low-energy spectra, correlation functions and the specific heat, agree qualitatively with the results obtained for finite samples of the kagom\'e AF. As in the case of the kagom\'e AF, these finite size results suggest that our model is a spin liquid with a gap between the ground state and the lowest spin excitation. However, extrapolations from DMRG data for strips of up to $N=120$ sites point towards a vanishing spin gap in a wide range of couplings, $0.5\ensuremath{\lesssim}{J/J}^{\ensuremath{'}}\ensuremath{\lesssim}1.25,$ so that contrary to the above conjecture, our one-dimensional model may in fact be critical in this parameter range. Moreover, our DMRG data indicate that the model undergoes a transition to a gapped state as the ratio ${J/J}^{\ensuremath{'}}$ increases through the value ${J/J}^{\ensuremath{'}}\ensuremath{\simeq}1.25.$ In the vicinity of the transition point, a high density of low-lying singlets develops in the spin gap.

Physical review. B, Condensed matter•Oct 1, 2000•35 citations

Correlated spin liquids in the quantum kagome antiferromagnet at finite field: a renormalization group analysis

Santanu Pal, Anirban Mukherjee, Siddhartha Lal

We analyze the antiferromagnetic spin-1/2 XXZ model on the kagome lattice at finite external magnetic field with the help of a non-perturbative zero-temperature renormalization group (RG) technique. The exact nature of the ground and excited state properties (e.g. gapped or gapless spectrum etc) of this system are still debated. Approximate methods have typically been adopted towards understanding the low-energy spectrum. Following the work of Kumar et al (2014 Phys. Rev. B 90 174409), we use a Jordan–Wigner transformation to map the spin problem into one of spinless fermions (spinons) in the presence of a statistical gauge field, and with nearest-neighbor interactions. While the work of Kumar et al was confined mostly to the plateau at 1/3-filling (magnetization per site) in the XY regime, we analyze the role of inter-spinon interactions in shaping the phases around this plateau in the entire XXZ model. The RG phase diagram obtained contains three spin liquid phases whose position is determined as a function of the exchange anisotropy and the energy scale for fluctuations arising from spinon scattering. Two of these spins liquids are topologically ordered states of matter with gapped, degenerate states on the torus. The gap for one of these phases corresponds to the one-spinon band gap of the Azbel–Hofstadter spectrum for the XY part of the Hamiltonian, while the other arises from two-spinon interactions. The Heisenberg point of this problem is found to lie within the interaction gapped spin liquid phase, in broad agreement with a recent experimental finding. The third phase is an algebraic spin liquid with a gapless Dirac spectrum for spinon excitations, and possess properties that show departures from the Fermi liquid paradigm. The three phase boundaries correspond to critical theories, and meet at a SU (2)-symmetric multicritical point. This special critical point agrees well with the gap-closing transition point predicted by Kumar et al . We discuss the relevance of our findings to various recent experiments, as well as results obtained from other theoretical analyses.

arXiv:1811.05222•New Journal of Physics•Feb 11, 2019•15 citations

Low-energy excitations of the kagomé antiferromagnet and the spin-gap issue

Philippe Sindzingre, C. Lhuillier

In this paper we report the latest results of exact diagonalizations of SU (2) invariant models on various lattices (square, triangular, hexagonal, checkerboard and kagomé lattices) and revisit critically in this light some of our previous analysis. We focus on the low-lying levels in each S sector. The differences in behavior between gapless systems and gapped ones are exhibited. The plausibility of a gapless spin liquid in the Heisenberg model on the kagomé lattice is discussed. A rough estimate of the spin susceptibility in such an hypothesis is given. The evolution of the intra- S -channel spectra under the effect of a small perturbation is consistent with the proximity of a quantum critical point. We reemphasize that the very small intra- S -channel energy scale observed in exact spectra is a very interesting information to understand the low- T dynamics of this model.

arXiv:0907.4164•Europhysics Letters (EPL)•Oct 1, 2009•106 citations
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′{J}^{\ensuremath{'}}J\ensuremath′, so that J\ensuremath′=J{J}^{\ensuremath{'}}=JJ\ensuremath′=J corresponds to the triangular limit and J\ensuremath′=0{J}^{\ensuremath{'}}=0J\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{J}^{\ensuremath{'}}∕JJ\ensuremath′∕J. Our results indicate that the antiferromagnetic order is not affected by the reduction of J\ensuremath′{J}^{\ensuremath{'}}J\ensuremath′ down to J\ensuremath′∕J\ensuremath≃0.2{J}^{\ensuremath{'}}∕J\ensuremath{\simeq}0.2J\ensuremath′∕J\ensuremath≃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→0{J}^{\ensuremath{'}}\ensuremath{\rightarrow}0J\ensuremath′\ensuremath→0 to the kagome limit. In particular, for any nonzero value of J\ensuremath′{J}^{\ensuremath{'}}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 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 5959%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.
A spin-1 Heisenberg model on trimerized kagome lattice is studied by doing a low-energy bosonic theory in terms of plaquette triplons defined on its triangular unit cells. The model considered has an intratriangle antiferromagnetic exchange interaction JJJ (set to 1) and two intertriangle couplings ${J}^{\ensuremath{'}}>0$ (nearest neighbor) and ${J}^{\ensuremath{''}}$ (next nearest neighbor; of both signs). The triplon analysis performed on this model investigates the stability of the trimerized singlet ground state (which is exact in the absence of intertriangle couplings) in the ${J}^{\ensuremath{'}}\ensuremath{-}{J}^{\ensuremath{''}}$ plane. It gives a quantum phase diagram that has two gapless antiferromagnetically ordered phases separated by the spin-gapped trimerized singlet phase. The trimerized singlet ground state is found to be stable on ${J}^{\ensuremath{''}}=0$ line (the nearest-neighbor case), and on both sides of it for ${J}^{\ensuremath{''}}\ensuremath{\ne}0$, in an extended region bounded by the critical lines of transition to the gapless antiferromagnetic phases. The gapless phase in the negative ${J}^{\ensuremath{''}}$ region has a coplanar ${120}^{\ensuremath{\circ}}$ antiferromagnetic order with $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ structure. In this phase, all the magnetic moments are of equal length, and the angle between any two of them on a triangle is exactly ${120}^{\ensuremath{\circ}}$. The magnetic lattice in this case has a unit cell consisting of three triangles. The other gapless phase, in the positive ${J}^{\ensuremath{''}}$ region, is found to exhibit a different coplanar antiferromagnetic order with ordering wave vector q=(0,0)\mathbf{q}=(0,0)q=(0,0). Here, two magnetic moments in a triangle are of the same magnitude, but shorter than the third. While the angle between two short moments is ${120}^{\ensuremath{\circ}}\ensuremath{-}2\ensuremath{\delta}$, it is ${120}^{\ensuremath{\circ}}+\ensuremath{\delta}$ between a short and the long one. Only when ${J}^{\ensuremath{''}}={J}^{\ensuremath{'}}$, their magnitudes become equal and the relative angles ${120}^{\ensuremath{\circ}}$. The magnetic lattice in this q=(0,0)\mathbf{q}=(0,0)q=(0,0) phase has the translational symmetry of the kagome lattice with triangular unit cells of reduced (isosceles) symmetry. This reduction in the point-group symmetry is found to show up as a difference in the intensities of certain Bragg peaks, whose ratio ${I}_{(1,0)}/{I}_{(0,1)}=4{sin}^{2}(\frac{\ensuremath{\pi}}{6}+\ensuremath{\delta})$ presents an experimental measure of the deviation $\ensuremath{\delta}$ from the ${120}^{\ensuremath{\circ}}$ order.
We have realized a thorough quantum mechanical analysis of an extensive number of low lying levels of the Heisenberg Hamiltonian on small periodic samples. Striking qualitative differences appear between the spectra of the triangular lattice and those of the Kagomé lattice. Symmetries and finite size scaling of the low lying levels of the triangular antiferromagnetic point to an à la Néel symmetry breaking, and the dynamical structure factors corroborate this conclusion. The Kagomé antiferromagnet spectra are qualitatively different: no signature of a symmetry breaking has been exhibited and the dynamical structure factors of the low lying levels display broad bands. The Heisenberg antiferromagnet on a Kagomé lattice is a serious candidate for a spin liquid.
We study a Heisenberg antiferromagnet (AF) on a one-dimensional strip composed of two rows of corner sharing triangular plaquettes. The geometry of this ladder-like lattice naturally allows for two different exchange couplings, one between pairs of spins on the outer legs (J) and a different one between the spins on the axis and their nearest neighbors on the legs ${(J}^{\ensuremath{'}}).$ For ${J/J}^{\ensuremath{'}}\ensuremath{\lesssim}0.5$ the model is a ferrimagnet. Our main interest is in the region ${J/J}^{\ensuremath{'}}\ensuremath{\gtrsim}0.5,$ where the classical ground state of this system shows the same macroscopic degeneracy as the classical ground state of the antiferromagnet on the kagom\'e lattice. To explore to which extent this similarity between the classical ground states of these two models carries over to their quantum states we have applied exact diagonalization techniques and density-matrix renormalization group (DMRG) methods to our model. Exact diagonalization is restricted to system sizes of up to N=30N=30N=30 sites. For ${J/J}^{\ensuremath{'}}=1,$ the results obtained by this technique, low-energy spectra, correlation functions and the specific heat, agree qualitatively with the results obtained for finite samples of the kagom\'e AF. As in the case of the kagom\'e AF, these finite size results suggest that our model is a spin liquid with a gap between the ground state and the lowest spin excitation. However, extrapolations from DMRG data for strips of up to N=120N=120N=120 sites point towards a vanishing spin gap in a wide range of couplings, $0.5\ensuremath{\lesssim}{J/J}^{\ensuremath{'}}\ensuremath{\lesssim}1.25,$ so that contrary to the above conjecture, our one-dimensional model may in fact be critical in this parameter range. Moreover, our DMRG data indicate that the model undergoes a transition to a gapped state as the ratio ${J/J}^{\ensuremath{'}}$ increases through the value ${J/J}^{\ensuremath{'}}\ensuremath{\simeq}1.25.$ In the vicinity of the transition point, a high density of low-lying singlets develops in the spin gap.
We analyze the antiferromagnetic spin-1/2 XXZ model on the kagome lattice at finite external magnetic field with the help of a non-perturbative zero-temperature renormalization group (RG) technique. The exact nature of the ground and excited state properties (e.g. gapped or gapless spectrum etc) of this system are still debated. Approximate methods have typically been adopted towards understanding the low-energy spectrum. Following the work of Kumar et al (2014 Phys. Rev. B 90 174409), we use a Jordan–Wigner transformation to map the spin problem into one of spinless fermions (spinons) in the presence of a statistical gauge field, and with nearest-neighbor interactions. While the work of Kumar et al was confined mostly to the plateau at 1/3-filling (magnetization per site) in the XY regime, we analyze the role of inter-spinon interactions in shaping the phases around this plateau in the entire XXZ model. The RG phase diagram obtained contains three spin liquid phases whose position is determined as a function of the exchange anisotropy and the energy scale for fluctuations arising from spinon scattering. Two of these spins liquids are topologically ordered states of matter with gapped, degenerate states on the torus. The gap for one of these phases corresponds to the one-spinon band gap of the Azbel–Hofstadter spectrum for the XY part of the Hamiltonian, while the other arises from two-spinon interactions. The Heisenberg point of this problem is found to lie within the interaction gapped spin liquid phase, in broad agreement with a recent experimental finding. The third phase is an algebraic spin liquid with a gapless Dirac spectrum for spinon excitations, and possess properties that show departures from the Fermi liquid paradigm. The three phase boundaries correspond to critical theories, and meet at a SU (2)-symmetric multicritical point. This special critical point agrees well with the gap-closing transition point predicted by Kumar et al . We discuss the relevance of our findings to various recent experiments, as well as results obtained from other theoretical analyses.
In this paper we report the latest results of exact diagonalizations of SU (2) invariant models on various lattices (square, triangular, hexagonal, checkerboard and kagomé lattices) and revisit critically in this light some of our previous analysis. We focus on the low-lying levels in each S sector. The differences in behavior between gapless systems and gapped ones are exhibited. The plausibility of a gapless spin liquid in the Heisenberg model on the kagomé lattice is discussed. A rough estimate of the spin susceptibility in such an hypothesis is given. The evolution of the intra- S -channel spectra under the effect of a small perturbation is consistent with the proximity of a quantum critical point. We reemphasize that the very small intra- S -channel energy scale observed in exact spectra is a very interesting information to understand the low- T dynamics of this model.