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

Physical review. B, Condensed matter

Effects of cyclic four-spin exchange on the magnetic properties of the CuO2 plane

Y. Honda, Yoshio Kuramoto, Tsukasa Watanabe

Published May 1, 1993• 66 citations
Publisher

Abstract

For the ${\mathrm{CuO}}_{2}$ square lattice, as a parent system of high-${\mathit{T}}_{\mathit{c}}$ superconductors, a numerical study has suggested that the effective-spin model contains a large cyclic four-spin exchange interaction ${\mathit{J}}_{\mathit{c}}$. This paper investigates the effects of ${\mathit{J}}_{\mathit{c}}$ upon magnetic Raman scattering and upon properties of the ground state by an exact numerical method mainly for a 16-site cluster. It is found that the main Raman peak with the ${\mathit{B}}_{1\mathit{g}}$ symmetry is shifted by 20% to the lower-energy side with a realistic magnitude for ${\mathit{J}}_{\mathit{c}}$, which is about one fourth the size of the nearest-neighbor exchange J. Accordingly a value of J extracted from analyses of experimental data becomes larger by 10% than that estimated with the use of the Heisenberg model. Furthermore the four-spin exchange enhances a shoulder due to multimagnon states at an energy of about 4J, and this result compares favorably with the experimental line shape. Detailed discussion is given on the effect of the four-spin exchange upon the ground-state properties such as the staggered magnetization, the ground-state energy, weight of various spin configurations, and the spin-spin correlation function.

DOI
10.1103/physrevb.47.11329

Related papers

Magnetic Raman Scattering in Two-Dimensional Spin-1/2 Heisenberg Antiferromagnets: Spectral Shape Anomaly and Magnetostrictive Effects

Franco Nori, Roberto D. Merlin, Stephan Haas +2 more

We calculate the Raman spectrum of the two-dimensional spin-1/2 Heisenberg antiferromagnet by exact diagonalization and quantum Monte Carlo techniques on clusters of up to 144 sites. On a 16-site cluster, we consider the phonon-magnon interaction which leads to random fluctuations of the exchange integral. Results are in good agreement with Raman scattering experiments on various high- ${T}_{c}$ precursors, such as ${\mathrm{La}}_{2}{\mathrm{CuO}}_{4}$ and ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6.2}$. In particular, our calculations reproduce the broad line shape of the two-magnon peak, the asymmetry about its maximum, the existence of spectral weight at high energies, and the observation of nominally forbidden ${A}_{1g}$ scattering.

arXiv:cond-mat/9505012•Physical Review Letters•Jul 17, 1995•42 citations

Dynamical Studies of Antiferromagnetic Exchange Interactions in Low Dimensional Quantum Spin Systems

Various forms of antiferromagnetic exchange interaction among quantized spins in one-dimensional and quasi-one-dimensional lattices are examined. Primary results are reported in two real compounds, the sodium vanadate NaV2O5 and the copper germanate CuGeO3, and in a class of compounds of real materials modeled as dimerized antiferromagnetically coupled spin-1/2 chains susceptible to spin-1 impurity doping and that include such examples as the strontium cuprate Sr14Cu24O41 and vanadium pyrophosphate (VO)2P2O7. Sodium vanadate, NaV2O5, a quarter-filled two-leg ladder compound that was originally thought to be composed of magnetic legs and nonmagnetic legs was subsequently shown to be a charge ordered system below room temperature. Initial models of the material as isolated antiferromagnetically coupled spin-1/2 chains can be mapped to the subsequently resolved magnetic system of the charge ordered state with the spins on molecular orbitals of V-O-V rungs antiferromagentically coupled between neighboring rungs either on the same ladder or on neighboring ladders. Comparing the model's dynamic structure factor to inelastic neutron scattering data could distinguish between the proposed spin coupling schemes and consequently reveal the details of the interaction between the lattice and the "zig-zag" charge density as well as the magnetic stabilization of the low temperature phase. Copper germanate, the first and only known inorganic spin-Peierls compound, exhibits a strong temperature dependent exchange coupling pattern among the spin-1/2 Cu2+ chains. Above the spin-Peierls transition temperature an unmodulated antiferromagnetic nearest neighbor exchange interaction J1 competes with an antiferromagnetic next nearest neighbor interaction J2. The next nearest neighbor exchange represents an effective coupling equivalent to the net effect of all longer range exchange interactions. This frustration to the nearest neighbor spin exchange produces a distinctive magnetic susceptibility χ (T) much different from the Bonner-Fisher susceptibility of the spin-1/2 Heisenberg chain with nearest neighbor antiferromagnetic exchange only. The ratio of the second to first nearest neighbor couplings α = J2/J1 ≈ 0.36 is sufficient to open a spontaneous gap in the spin-wave excitation spectrum at low temperatures. Below TSP = 14K, the dynamical structure factor is used to fit the dimerization δ and the exchange interactions J1 and J2 to the inelastic neutron scattering data of CuGeO3 at T = 10K. It is found that both δ and α increase significantly at lower temperatures, relative to the values obtained in the high temperature phase and at the onset of dimeration at 14 K. Static structure factor calculations how Scattering inconsistent with the δ−J1−J2 model and can be attributed primarily to the phonon degrees of freedom but possibly also to the couplings between chains in the b and a directions of the crystal lattice. Structurally dimerized compounds of antiferromagnetic spin-1/2 chains possess dynamic structure factor and magnetic susceptibility features that are very sensitive to doping with magnetic and nonmagnetic impurities. It is shown that the effects of spin-1 impurities are very similar to those of nonmagnetic (S = 0) impurities if the coupling between impurity and native spin is about the same magnitude as or larger than between native S = 1/2 spins. The microscopic origins of the similarity can be appreciated with the consideration that neighboring spins to the impurity find it more energetically favorable to couple to the impurity than the other spins of the lattice. Thus the spin-1 impurity and its neighboring S = 1/2 spins decouple from the lattice, creating a nonmagnetic break in the chain. The spins that now observe a nonmagnetic cluster on one side can couple to the S = 1/2 spins on the other side with a higher AF correlation than in the pure compound since they do not have to have a fluctuating dimer resonating between two neighbors. Such enhancements to the local correlations can stabilize the global AF order in one dimensional compounds as shown in theoretical and experimental studies of CuGeO3. ED calculations of the dynamic structure factor show that S(q, ω) developes states with ω = 0 in the gap at q = π. Magnetic susceptibilites in the presense of spin-1 impurities were obtained for 80 site chains by QMC simulations and for 16 site chains by ED calculations. An impurity concentration-dependent second peak in the low temperature region reveals an increasing second maximum with T as observed in doped copper germanates and suggests that the results of the study are applicable even to phonon mediated dimerization. This is not immediately apparent because spin-phonon coupling could distort the dimerization patterns. However, experimental and theoretical results show that elastic coupling between chains can stabilize the dimerization pattern due to the impurities and the results will still apply.

•Jan 1, 2003•0 citations

Signatures of next-nearest-neighbor (NNN) Antiferromagnetic Exchange Interaction on High-Tc Superconductivity Within t-t'-J-J' Model

K Roy, Papiya Pal, Nanda Kumar Ghosh

The influence of the next-nearest-neighbor (NNN) antiferromagnetic interaction J ’has been examined within t-t'-J-J ’model in an exact technique. An 8-site square cluster is chosen as the representative of the system. The effect of the next-nearest-neighbor (NNN) antiferromagnetic interaction on various ground state properties such as electron-electron correlation, spin-spin correlation, effective hopping amplitude etc. has been studied. Temperature variation of the susceptibility shows that J ’suppresses short range antiferromagnetic order. Applying periodic boundary conditions, the calculations have been extended beyond the 8-site limit.

Bulletin of Pure & Applied Sciences- Physics•Jan 1, 2018•2 citations

Ring exchange and the Heisenberg and Hubbard models

M W Long, C. W. M. Castleton, C A Hayward

We study ring-exchange or cyclic-permutation correlations in one-dimensional quantum spin-half systems. For the Heisenberg model we show numerically that these correlations decay as R n approximately 1/ square root n, although we can deduce nothing about any possible important logarithmic corrections. As such, ring-exchange correlations are much longer range than the more commonly considered spin-spin correlation functions. By considering the relationship between solitonic excitations and cyclic permutations, we suggest a way to predict the value of J 2 /J 1 at which the phase transition between a gapped and gapless phase occurs in the next-nearest-neighbour Heisenberg model, suggesting J 2 =4J 1 as the exact transition point. For the Hubbard model with a spin-charge-separated solution, we show that the occupation number, n k , is a 'convolution' of the cyclic-permutation correlations of the spin ground state with the anyonic occupation number of the charge ground state, with the integration being over statistical phase. We deduce the one-eighth singularity previously found for the U= infinity Hubbard model using this new route. We show that for the limit where nearest-neighbour hopping dominates longer-range hopping in the U= infinity Hubbard model, the single-particle correlation function, n k , for an infinitesimal concentration of holes in a half-filled system, is identical to the Fourier transform of the cyclic-exchange correlations of the corresponding spin wavefunction. For the elementary t 1 -t 2 model, we show a relationship between the singularity which occurs at the Fermi surface, the so-called Luttinger-liquid singularity, and the long-range Heisenberg-model cyclic-permutation correlations.

Journal of Physics Condensed Matter•Oct 31, 1994•2 citations

Exact diagonalization study of the triangular Heisenberg model with four-spin ring-exchange interaction

Yu Zheng, Muwei Wu, Dao‐Xin Yao +1 more

Using Lanczos exact diagonalization (ED), we study the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model with the four-spin ring-exchange interaction ${J}_{r}$ on a triangular lattice. We mainly use the level spectroscopic technique of two 36-site tori to investigate the ground-state phase diagram, and further characterize phases by spin, dimer, and chiral correlation functions. The ground state has rich phases including several magnetic-ordered phases like the zigzag phase and tetrahedral phase, as well as several novel nonmagnetic phases, some of which exhibit valence bond solid behavior in their dimer correlation functions. However, we do not find direct evidence of a quantum spin liquid phase with spinon Fermi surface in this model. Our results can give a better understanding of the ground-state properties of the triangular Heisenberg model with ring-exchange interaction and help to understand the relevant triangular materials.

arXiv:2505.02030•Physical review. B./Physical review. B•May 21, 2025•1 citation

Spin- 12 J1 - J2 Heisenberg model on a cross-striped square lattice

R. F. Bishop, P. H. Y. Li, C. E. Campbell

Using the coupled cluster method (CCM) we study the full (zero-temperature) ground-state (GS) phase diagram of a spin-half ($s=\frac{1}{2}$) ${J}_{1}$-${J}_{2}$ Heisenberg model on a cross-striped square lattice. Each site of the square lattice has four nearest-neighbor exchange bonds of strength ${J}_{1}$ and two next-nearest-neighbor (diagonal) bonds of strength ${J}_{2}$. The ${J}_{2}$ bonds are arranged so that the basic square plaquettes in alternating columns have either both or no ${J}_{2}$ bonds included. The classical ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) version of the model has four collinear phases when ${J}_{1}$ and ${J}_{2}$ can take either sign. Three phases are antiferromagnetic (AFM), showing so-called N\'eel, double N\'eel, and double columnar striped order, respectively, while the fourth is ferromagnetic. For the quantum $s=\frac{1}{2}$ model we use the three classical AFM phases as CCM reference states, on top of which the multispin-flip configurations arising from quantum fluctuations are incorporated in a systematic truncation hierarchy. Calculations of the corresponding GS energy, magnetic order parameter, and the susceptibilities of the states to various forms of valence-bond crystalline (VBC) order are thus carried out numerically to high orders of approximation and then extrapolated to the (exact) physical limit. We find that the $s=\frac{1}{2}$ model has five phases, which correspond to the four classical phases plus a new quantum phase with plaquette VBC order. The positions of the five quantum critical points are determined with high accuracy. While all four phase transitions in the classical model are first order, we find strong evidence that three of the five quantum phase transitions in the $s=\frac{1}{2}$ model are of continuous deconfined type.

arXiv:1308.4573•Physical Review B•Dec 19, 2013•10 citations
For the CuO2{\mathrm{CuO}}_{2}CuO2​ square lattice, as a parent system of high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors, a numerical study has suggested that the effective-spin model contains a large cyclic four-spin exchange interaction Jc{\mathit{J}}_{\mathit{c}}Jc​. This paper investigates the effects of Jc{\mathit{J}}_{\mathit{c}}Jc​ upon magnetic Raman scattering and upon properties of the ground state by an exact numerical method mainly for a 16-site cluster. It is found that the main Raman peak with the B1g{\mathit{B}}_{1\mathit{g}}B1g​ symmetry is shifted by 20% to the lower-energy side with a realistic magnitude for Jc{\mathit{J}}_{\mathit{c}}Jc​, which is about one fourth the size of the nearest-neighbor exchange J. Accordingly a value of J extracted from analyses of experimental data becomes larger by 10% than that estimated with the use of the Heisenberg model. Furthermore the four-spin exchange enhances a shoulder due to multimagnon states at an energy of about 4J, and this result compares favorably with the experimental line shape. Detailed discussion is given on the effect of the four-spin exchange upon the ground-state properties such as the staggered magnetization, the ground-state energy, weight of various spin configurations, and the spin-spin correlation function.
We calculate the Raman spectrum of the two-dimensional spin-1/2 Heisenberg antiferromagnet by exact diagonalization and quantum Monte Carlo techniques on clusters of up to 144 sites. On a 16-site cluster, we consider the phonon-magnon interaction which leads to random fluctuations of the exchange integral. Results are in good agreement with Raman scattering experiments on various high- Tc{T}_{c}Tc​ precursors, such as La2CuO4{\mathrm{La}}_{2}{\mathrm{CuO}}_{4}La2​CuO4​ and YBa2Cu3O6.2{\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6.2}YBa2​Cu3​O6.2​. In particular, our calculations reproduce the broad line shape of the two-magnon peak, the asymmetry about its maximum, the existence of spectral weight at high energies, and the observation of nominally forbidden A1g{A}_{1g}A1g​ scattering.
Various forms of antiferromagnetic exchange interaction among quantized spins in one-dimensional and quasi-one-dimensional lattices are examined. Primary results are reported in two real compounds, the sodium vanadate NaV2O5 and the copper germanate CuGeO3, and in a class of compounds of real materials modeled as dimerized antiferromagnetically coupled spin-1/2 chains susceptible to spin-1 impurity doping and that include such examples as the strontium cuprate Sr14Cu24O41 and vanadium pyrophosphate (VO)2P2O7. Sodium vanadate, NaV2O5, a quarter-filled two-leg ladder compound that was originally thought to be composed of magnetic legs and nonmagnetic legs was subsequently shown to be a charge ordered system below room temperature. Initial models of the material as isolated antiferromagnetically coupled spin-1/2 chains can be mapped to the subsequently resolved magnetic system of the charge ordered state with the spins on molecular orbitals of V-O-V rungs antiferromagentically coupled between neighboring rungs either on the same ladder or on neighboring ladders. Comparing the model's dynamic structure factor to inelastic neutron scattering data could distinguish between the proposed spin coupling schemes and consequently reveal the details of the interaction between the lattice and the "zig-zag" charge density as well as the magnetic stabilization of the low temperature phase. Copper germanate, the first and only known inorganic spin-Peierls compound, exhibits a strong temperature dependent exchange coupling pattern among the spin-1/2 Cu2+ chains. Above the spin-Peierls transition temperature an unmodulated antiferromagnetic nearest neighbor exchange interaction J1 competes with an antiferromagnetic next nearest neighbor interaction J2. The next nearest neighbor exchange represents an effective coupling equivalent to the net effect of all longer range exchange interactions. This frustration to the nearest neighbor spin exchange produces a distinctive magnetic susceptibility χ (T) much different from the Bonner-Fisher susceptibility of the spin-1/2 Heisenberg chain with nearest neighbor antiferromagnetic exchange only. The ratio of the second to first nearest neighbor couplings α = J2/J1 ≈ 0.36 is sufficient to open a spontaneous gap in the spin-wave excitation spectrum at low temperatures. Below TSP = 14K, the dynamical structure factor is used to fit the dimerization δ and the exchange interactions J1 and J2 to the inelastic neutron scattering data of CuGeO3 at T = 10K. It is found that both δ and α increase significantly at lower temperatures, relative to the values obtained in the high temperature phase and at the onset of dimeration at 14 K. Static structure factor calculations how Scattering inconsistent with the δ−J1−J2 model and can be attributed primarily to the phonon degrees of freedom but possibly also to the couplings between chains in the b and a directions of the crystal lattice. Structurally dimerized compounds of antiferromagnetic spin-1/2 chains possess dynamic structure factor and magnetic susceptibility features that are very sensitive to doping with magnetic and nonmagnetic impurities. It is shown that the effects of spin-1 impurities are very similar to those of nonmagnetic (S = 0) impurities if the coupling between impurity and native spin is about the same magnitude as or larger than between native S = 1/2 spins. The microscopic origins of the similarity can be appreciated with the consideration that neighboring spins to the impurity find it more energetically favorable to couple to the impurity than the other spins of the lattice. Thus the spin-1 impurity and its neighboring S = 1/2 spins decouple from the lattice, creating a nonmagnetic break in the chain. The spins that now observe a nonmagnetic cluster on one side can couple to the S = 1/2 spins on the other side with a higher AF correlation than in the pure compound since they do not have to have a fluctuating dimer resonating between two neighbors. Such enhancements to the local correlations can stabilize the global AF order in one dimensional compounds as shown in theoretical and experimental studies of CuGeO3. ED calculations of the dynamic structure factor show that S(q, ω) developes states with ω = 0 in the gap at q = π. Magnetic susceptibilites in the presense of spin-1 impurities were obtained for 80 site chains by QMC simulations and for 16 site chains by ED calculations. An impurity concentration-dependent second peak in the low temperature region reveals an increasing second maximum with T as observed in doped copper germanates and suggests that the results of the study are applicable even to phonon mediated dimerization. This is not immediately apparent because spin-phonon coupling could distort the dimerization patterns. However, experimental and theoretical results show that elastic coupling between chains can stabilize the dimerization pattern due to the impurities and the results will still apply.
The influence of the next-nearest-neighbor (NNN) antiferromagnetic interaction J ’has been examined within t-t'-J-J ’model in an exact technique. An 8-site square cluster is chosen as the representative of the system. The effect of the next-nearest-neighbor (NNN) antiferromagnetic interaction on various ground state properties such as electron-electron correlation, spin-spin correlation, effective hopping amplitude etc. has been studied. Temperature variation of the susceptibility shows that J ’suppresses short range antiferromagnetic order. Applying periodic boundary conditions, the calculations have been extended beyond the 8-site limit.
We study ring-exchange or cyclic-permutation correlations in one-dimensional quantum spin-half systems. For the Heisenberg model we show numerically that these correlations decay as R n approximately 1/ square root n, although we can deduce nothing about any possible important logarithmic corrections. As such, ring-exchange correlations are much longer range than the more commonly considered spin-spin correlation functions. By considering the relationship between solitonic excitations and cyclic permutations, we suggest a way to predict the value of J 2 /J 1 at which the phase transition between a gapped and gapless phase occurs in the next-nearest-neighbour Heisenberg model, suggesting J 2 =4J 1 as the exact transition point. For the Hubbard model with a spin-charge-separated solution, we show that the occupation number, n k , is a 'convolution' of the cyclic-permutation correlations of the spin ground state with the anyonic occupation number of the charge ground state, with the integration being over statistical phase. We deduce the one-eighth singularity previously found for the U= infinity Hubbard model using this new route. We show that for the limit where nearest-neighbour hopping dominates longer-range hopping in the U= infinity Hubbard model, the single-particle correlation function, n k , for an infinitesimal concentration of holes in a half-filled system, is identical to the Fourier transform of the cyclic-exchange correlations of the corresponding spin wavefunction. For the elementary t 1 -t 2 model, we show a relationship between the singularity which occurs at the Fermi surface, the so-called Luttinger-liquid singularity, and the long-range Heisenberg-model cyclic-permutation correlations.
Using Lanczos exact diagonalization (ED), we study the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model with the four-spin ring-exchange interaction Jr{J}_{r}Jr​ on a triangular lattice. We mainly use the level spectroscopic technique of two 36-site tori to investigate the ground-state phase diagram, and further characterize phases by spin, dimer, and chiral correlation functions. The ground state has rich phases including several magnetic-ordered phases like the zigzag phase and tetrahedral phase, as well as several novel nonmagnetic phases, some of which exhibit valence bond solid behavior in their dimer correlation functions. However, we do not find direct evidence of a quantum spin liquid phase with spinon Fermi surface in this model. Our results can give a better understanding of the ground-state properties of the triangular Heisenberg model with ring-exchange interaction and help to understand the relevant triangular materials.
Using the coupled cluster method (CCM) we study the full (zero-temperature) ground-state (GS) phase diagram of a spin-half (s=12s=\frac{1}{2}s=21​) J1{J}_{1}J1​-J2{J}_{2}J2​ Heisenberg model on a cross-striped square lattice. Each site of the square lattice has four nearest-neighbor exchange bonds of strength J1{J}_{1}J1​ and two next-nearest-neighbor (diagonal) bonds of strength J2{J}_{2}J2​. The J2{J}_{2}J2​ bonds are arranged so that the basic square plaquettes in alternating columns have either both or no J2{J}_{2}J2​ bonds included. The classical ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) version of the model has four collinear phases when J1{J}_{1}J1​ and J2{J}_{2}J2​ can take either sign. Three phases are antiferromagnetic (AFM), showing so-called N\'eel, double N\'eel, and double columnar striped order, respectively, while the fourth is ferromagnetic. For the quantum s=12s=\frac{1}{2}s=21​ model we use the three classical AFM phases as CCM reference states, on top of which the multispin-flip configurations arising from quantum fluctuations are incorporated in a systematic truncation hierarchy. Calculations of the corresponding GS energy, magnetic order parameter, and the susceptibilities of the states to various forms of valence-bond crystalline (VBC) order are thus carried out numerically to high orders of approximation and then extrapolated to the (exact) physical limit. We find that the s=12s=\frac{1}{2}s=21​ model has five phases, which correspond to the four classical phases plus a new quantum phase with plaquette VBC order. The positions of the five quantum critical points are determined with high accuracy. While all four phase transitions in the classical model are first order, we find strong evidence that three of the five quantum phase transitions in the s=12s=\frac{1}{2}s=21​ model are of continuous deconfined type.