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:1103.5829 · Journal of the Physical Society of Japan

Numerical-Diagonalization Study of Spin Gap Issue of the Kagome Lattice Heisenberg Antiferromagnet

Hiroki Nakano, Tôru Sakai

Published May 11, 2011• 149 citations
arXivHTMLPDF

Abstract

We study the system size dependence of the singlet-triplet excitation gap in the $S=1/2$ kagome-lattice Heisenberg antiferromagnet by numerical diagonalization. We successfully obtain a new result of a cluster of 42 sites. The two sequences of gaps of systems with even-number sites and that with odd-number sites are separately analyzed. Careful examination clarifies that there is no contradiction when we consider the system to be gapless.

DOI
10.1143/jpsj.80.053704

Related papers

Density Matrix Renormalization Group Numerical Study of the Kagome Antiferromagnet

Hong‐Chen Jiang, Zheng-Yu Weng, D. N. Sheng

We numerically study the spin-1/2 antiferromagnetic Heisenberg model on the kagome lattice using the density-matrix renormalization group method. We find that the ground state is a magnetically disordered spin liquid, characterized by an exponential decay of spin-spin correlation function in real space and a magnetic structure factor showing system-size independent peaks at commensurate magnetic wave vectors. We obtain a spin triplet excitation gap DeltaE(S=1)=0.055+/-0.005 by extrapolation based on the large size results, and confirm the presence of gapless singlet excitations. The physical nature of such an exotic spin liquid is also discussed.

Physical Review Letters•Sep 10, 2008•271 citations

Triplet and singlet excitations in the valence bond crystal phase of the kagome lattice Heisenberg model

Rajiv R. P. Singh, David A. Huse

A proposed ground state for the kagome lattice Heisenberg model consists of a valence bond crystal (VBC) with a 36-site unit cell. We calculate the low-lying triplet and singlet excitations in the VBC phase for the infinite-lattice model and for the 36-site cluster. For the infinite lattice, the lowest triplet excitation is found to have a spin gap of approximately $0.08\ifmmode\pm\else\textpm\fi{}0.02J$ and a bandwidth of only about $0.01J$. For the 36-site cluster, which consists of a single unit cell with periodic boundary conditions, there are substantial finite-size effects: the spin gap there is estimated to be approximately $0.2J$, which is close to the exact diagonalization result of $0.164J$. The triplet excitations attract one another and form many bound states in the spin-singlet channel. We find a large number of such bound states for the 36-site cluster, many of which appear to lie below the spin gap, which is again in agreement with the results from exact diagonalization.

arXiv:0801.2735•Physical Review B•Apr 17, 2008•101 citations

DMRG Numerical Study of the Kagomé Antiferromagnet

Hong‐Chen Jiang, Zheng-Yu Weng, D. N. Sheng

We numerically study the spin-1/2 antiferromagnetic Heisenberg model on the kagomé lattice using the density-matrix renormalization group (DMRG) method. We find that the ground state is a magnetically disordered spin liquid, characterized by an exponential decay of spin-spin correlation function in real space and a magnetic structure factor showing system-size independent peaks at commersurate antiferromangetic wavevectors. We obtain a spin triplet excitation gap $ΔE(S=1)=0.055\pm 0.005$ by extrapolation based on the large size results, and confirm the presence of gapless singlet excitations. The physical nature of such an exotic spin liquid is also discussed.

arXiv:0804.1616•Apr 10, 2008•0 citations

Quantum Spin Liquid of Kagome-Lattice Antiferromagnet

Tôru Sakai, Tôru Sakai

The S=1/2 kagome-lattice antiferromagnet is one of interesting frustrated quantum spin systems. The system is supposed to exhibit the quantum spin liquid in the ground state, which was proposed as a possible origin of the high-temprature superconductivity. The spin gap is an important physical quantity to characterize the spin liquid behavior. Whether the S=1/2 kagome-lattice antiferromagnet is gapless or has a finite spin gap, is still an unsolved issue. Because any recently developped numerical calculation methods are not enough to determine it in the thermodynamic limit. Our large-scale numerical diazonalization up to 42-spin clusters and a finite-size scaling analysis indicated that the S=1/2 kagome-lattice antiferromagnet is gapless in the thremodynamic limit[1]. It is consistent with the U(1) Dirac spin liquid theory of the kagome-lattice antiferromagnet[2,3]. On the other hand, some density matrix renormalization group (DMRG) calculations supported the gapped Z2 topological spin liquid theory[4,5]. Our recent numerical diagonalization analysis on the magnetization process of a distorted kagome-lattice antiferromagnet indicated that the perfect kagome-lattice system is just on a quantum critical point[6]. It would be a possible reason why it is difficult to determine whether the perfect kagome-lattice antiferromagnet is gapless or gapped. In the present study, a field derivative analysis is applied for the spin gap issue of the kagome-lattice antiferromanget, using the numerical diagonalization data up to 42-spin clusters. It is consistent with our previous conclusion that it is gapless[7]. [1]H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 80 (2011) 053704. [2]Y. Ran, M. Hermele, P. A. Lee and X. -G. Wen: Phys. Rev. Lett. 98 (2007) 117205. [3]Y. Iqbal, F. Becca, S. Sorella and D. Poilblanc: Phys. Rev. B 87 (2013) 060405(R). [4]S. Yan, D. A. Huse and S. R. White: Science 332 (2011) 1173. [5]S. Nishimoto, N. Shibata and C. Hotta: Nat. Commun. 4 (2013) 2287. [6]H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 83 (2014) 104710. [7]H. Nakano and T. Sakai: in preparation.

•Nov 8, 2017•0 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

Study of Low-energy States of Clusters of Spin-1/2 and Spin-1 Triangles with Kagome-like Geometries

Indranil Rudra, S. Ramasesha, Diptiman Sen

We study the low-energy properties of Heisenberg antiferromagnetic spin-1/2 and spin-1 systems on various clusters made up of triangles. Some of the clusters have a geometry similar to representative pieces of the Kagome lattice, while others have the geometry of a sawtooth chain. While the ground state always has the lowest possible spin (0 or 1/2), the nature of the low-energy excitations depends on the geometry and the site spin. For the Kagome clusters with spin-1 sites, the lowest excitations are gapped, with singlet and triplet excitations having similar gaps. This is in contrast to Kagome clusters with spin-1/2 sites where there are several low-energy singlet excitations (possibly gapless in the thermodynamic limit), while triplet excitations have a gap. For the sawtooth chain with spin-1 sites, the lowest excitations are triplets with a gap; the gap to singlet excitations is about twice the triplet gap.

arXiv:cond-mat/0210122•Oct 6, 2002•0 citations
We study the system size dependence of the singlet-triplet excitation gap in the S=1/2S=1/2S=1/2 kagome-lattice Heisenberg antiferromagnet by numerical diagonalization. We successfully obtain a new result of a cluster of 42 sites. The two sequences of gaps of systems with even-number sites and that with odd-number sites are separately analyzed. Careful examination clarifies that there is no contradiction when we consider the system to be gapless.
We numerically study the spin-1/2 antiferromagnetic Heisenberg model on the kagome lattice using the density-matrix renormalization group method. We find that the ground state is a magnetically disordered spin liquid, characterized by an exponential decay of spin-spin correlation function in real space and a magnetic structure factor showing system-size independent peaks at commensurate magnetic wave vectors. We obtain a spin triplet excitation gap DeltaE(S=1)=0.055+/-0.005 by extrapolation based on the large size results, and confirm the presence of gapless singlet excitations. The physical nature of such an exotic spin liquid is also discussed.
A proposed ground state for the kagome lattice Heisenberg model consists of a valence bond crystal (VBC) with a 36-site unit cell. We calculate the low-lying triplet and singlet excitations in the VBC phase for the infinite-lattice model and for the 36-site cluster. For the infinite lattice, the lowest triplet excitation is found to have a spin gap of approximately $0.08\ifmmode\pm\else\textpm\fi{}0.02J$ and a bandwidth of only about 0.01J0.01J0.01J. For the 36-site cluster, which consists of a single unit cell with periodic boundary conditions, there are substantial finite-size effects: the spin gap there is estimated to be approximately 0.2J0.2J0.2J, which is close to the exact diagonalization result of 0.164J0.164J0.164J. The triplet excitations attract one another and form many bound states in the spin-singlet channel. We find a large number of such bound states for the 36-site cluster, many of which appear to lie below the spin gap, which is again in agreement with the results from exact diagonalization.
We numerically study the spin-1/2 antiferromagnetic Heisenberg model on the kagomé lattice using the density-matrix renormalization group (DMRG) method. We find that the ground state is a magnetically disordered spin liquid, characterized by an exponential decay of spin-spin correlation function in real space and a magnetic structure factor showing system-size independent peaks at commersurate antiferromangetic wavevectors. We obtain a spin triplet excitation gap ΔE(S=1)=0.055±0.005ΔE(S=1)=0.055\pm 0.005ΔE(S=1)=0.055±0.005 by extrapolation based on the large size results, and confirm the presence of gapless singlet excitations. The physical nature of such an exotic spin liquid is also discussed.
The S=1/2 kagome-lattice antiferromagnet is one of interesting frustrated quantum spin systems. The system is supposed to exhibit the quantum spin liquid in the ground state, which was proposed as a possible origin of the high-temprature superconductivity. The spin gap is an important physical quantity to characterize the spin liquid behavior. Whether the S=1/2 kagome-lattice antiferromagnet is gapless or has a finite spin gap, is still an unsolved issue. Because any recently developped numerical calculation methods are not enough to determine it in the thermodynamic limit. Our large-scale numerical diazonalization up to 42-spin clusters and a finite-size scaling analysis indicated that the S=1/2 kagome-lattice antiferromagnet is gapless in the thremodynamic limit[1]. It is consistent with the U(1) Dirac spin liquid theory of the kagome-lattice antiferromagnet[2,3]. On the other hand, some density matrix renormalization group (DMRG) calculations supported the gapped Z2 topological spin liquid theory[4,5]. Our recent numerical diagonalization analysis on the magnetization process of a distorted kagome-lattice antiferromagnet indicated that the perfect kagome-lattice system is just on a quantum critical point[6]. It would be a possible reason why it is difficult to determine whether the perfect kagome-lattice antiferromagnet is gapless or gapped. In the present study, a field derivative analysis is applied for the spin gap issue of the kagome-lattice antiferromanget, using the numerical diagonalization data up to 42-spin clusters. It is consistent with our previous conclusion that it is gapless[7]. [1]H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 80 (2011) 053704. [2]Y. Ran, M. Hermele, P. A. Lee and X. -G. Wen: Phys. Rev. Lett. 98 (2007) 117205. [3]Y. Iqbal, F. Becca, S. Sorella and D. Poilblanc: Phys. Rev. B 87 (2013) 060405(R). [4]S. Yan, D. A. Huse and S. R. White: Science 332 (2011) 1173. [5]S. Nishimoto, N. Shibata and C. Hotta: Nat. Commun. 4 (2013) 2287. [6]H. Nakano and T. Sakai: J. Phys. Soc. Jpn. 83 (2014) 104710. [7]H. Nakano and T. Sakai: in preparation.
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 the low-energy properties of Heisenberg antiferromagnetic spin-1/2 and spin-1 systems on various clusters made up of triangles. Some of the clusters have a geometry similar to representative pieces of the Kagome lattice, while others have the geometry of a sawtooth chain. While the ground state always has the lowest possible spin (0 or 1/2), the nature of the low-energy excitations depends on the geometry and the site spin. For the Kagome clusters with spin-1 sites, the lowest excitations are gapped, with singlet and triplet excitations having similar gaps. This is in contrast to Kagome clusters with spin-1/2 sites where there are several low-energy singlet excitations (possibly gapless in the thermodynamic limit), while triplet excitations have a gap. For the sawtooth chain with spin-1 sites, the lowest excitations are triplets with a gap; the gap to singlet excitations is about twice the triplet gap.