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./Physical review. B

Spectral evolution of the s=12 antiferromagnetic Heisenberg model: From one to two dimensions

Cheng Gu, Zhao-Long Gu, Shun-Li Yu, Jian‐Xin Li

Published Dec 13, 2023• 4 citations
Publisher

Abstract

We employ spin-cluster perturbation theory to investigate the evolution of the spin excitation spectra of the mixed-dimensional antiferromagnetic $s=\frac{1}{2}$ Heisenberg models from the one-dimensional chain to the two-dimensional square and honeycomb lattices by varying the spin interactions ${J}_{\ensuremath{\perp}}$ across the chains relative to ${J}_{\ensuremath{\parallel}}$ along the chains. In addition to the well-defined magnon excitations, the spectra of the N\'eel phases for both lattices exhibit anomalous high-energy continua that evolve continuously from the continuum of one-dimensional chain through increasing ${J}_{\ensuremath{\perp}}$. Notably, the key features of the continua at the two-dimensional limit for ${J}_{\ensuremath{\perp}}/{J}_{\ensuremath{\parallel}}=1$ are well consistent with experimental observations in real materials. Our results indicate that these high-energy continua in the N\'eel phases are a result of spinon deconfinement, which is further supported by a random-phase-approximation analysis based on the spinon Hubbard models.

DOI
10.1103/physrevb.108.224418

Related papers

Spin dynamics and continuum spectra of the honeycomb J 1 − J 2 antiferromagnetic Heisenberg model

Cheng Gu, Shun-Li Yu, Jian‐Xin Li

We employ the spin cluster perturbation theory to investigate the dynamical properties of the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the honeycomb lattice. We obtain the excitation spectra for all possible phases in the phase diagram, including the N\'eel phase, plaquette valence-bond-solid phase, dimer valence-bond-solid phase, and stripe antiferromagnetic phase. In the N\'eel phase, besides the obvious renormalization of the magnon dispersion, we find that the spectrum exhibits a dome-shaped broad continuum around the second Brillouin zone (BZ) and the additional strong continuum close to the corner of the BZ. In the valence-bond-solid phases, the spectra are dominated by a strong broad continuum all the way down to below ${J}_{1}$ coexisting with the lowest-energy triplon modes characterizing the plaquette and dimer phases. We ascribe this strong broad continuum and the additional continuum close to the BZ corner in the N\'eel phase to the contributions of fractionalized spinon excitations. In the stripe phase, a clear difference from the linear-spin-wave approximation is that the spectrum is gapped at the $M$ point, while that obtained by the latter is gapless due to the strong quantum fluctuations. We point out that the features observed in the N\'eel phase are consistent with the recent neutron scattering experiments on ${\mathrm{YbCl}}_{3}$ and ${\mathrm{YbBr}}_{3}$.

arXiv:2205.12822•Physical review. B./Physical review. B•May 9, 2022•10 citations

Spin excitation spectra of two dimensional $S=1/2$ Heisenberg model with checkerboard structures.

Yining Xu, Zijian Xiong, Han-Qing Wu +1 more

We study the excitation spectra of two-dimensional spin-$1/2$ Heisenberg model with checkerboard structures using stochastic analytic continuation of imaginary-time correlation function obtained from quantum Monte Carlo simulation. The checkerboard models have two different antiferromagnetic nearest-neighbor interactions $J_{1}$ and $J_{2}$, and the tuning parameter $g$ is defined as $J_{2}/J_{1}$. The dynamic structure factors are systematically calculated in all phases of the models as well as at the critical points. To give a full understanding of the dynamic spectra, wave theory is employed to explain some features of numerical results, especially for the low-energy part. When $g$ is close to $1$, the features of the excitation spectra of each checkerboard model are roughly the same as those of the original square lattice antiferromagnetic Heisenberg model, and the high-energy continuum among them are discussed. In contrast to the other checkerboard structures investigated in this paper, the $3\times 3$ checkerboard model has distinctive excitation features, like a gap between a low-energy gapless branch and a gapped high-energy part exists when $g$ is small. The gapless branch in this case can be regarded as wave in N$\mathrm{\acute{e}}$el order formed by block spin in each $3\times 3$ plaquette with effective exchange interaction originated from renormalization. One finding is that the continuum also appears in this low-energy branch, which may originated from nearly deconfined process in block spin.

•Dec 2, 2018•0 citations

Spin excitation spectra of the two-dimensional S=12 Heisenberg model with a checkerboard structure

Yining Xu, Zijian Xiong, Han-Qing Wu +1 more

We study the spin excitation spectra of the two-dimensional spin-$\frac{1}{2}$ Heisenberg model with a checkerboard structure using stochastic analytic continuation of the imaginary-time correlation function obtained from a quantum Monte Carlo simulation. The checkerboard models have two different antiferromagnetic nearest-neighbor interactions ${J}_{1}$ and ${J}_{2}$, and the tuning parameter $g$ is defined as ${J}_{2}/{J}_{1}$. The dynamic spin structure factors are systematically calculated in all phases of the models as well as at the critical points. To give a full understanding of the dynamic spectra, spin wave theory is employed to explain some features of numerical results, especially for the low-energy part. When $g$ is close to 1, the features of the spin excitation spectra of each checkerboard model are roughly the same as those of the original square lattice antiferromagnetic Heisenberg model, and the high-energy continuum among them is discussed. In contrast to the other checkerboard structures investigated in this paper, the $3\ifmmode\times\else\texttimes\fi{}3$ checkerboard model has distinctive excitation features, such as a gap between a low-energy gapless branch and a gapped high-energy part that exists when $g$ is small. The gapless branch in this case can be regarded as a spin wave in N\'eel order formed by a ``block spin'' in each $3\ifmmode\times\else\texttimes\fi{}3$ plaquette with an effective exchange interaction originating from renormalization. One unexpected finding is that the continuum also appears in this low-energy branch.

arXiv:1811.12753•Physical review. B./Physical review. B•Feb 7, 2019•17 citations

Spin wave excitations and their interactions in two-dimensional antiferromagnets

Dag-Björn Hering

This thesis investigates two-dimensional quantum antiferromagnets, with a focus on the spin-$\frac{1}{2}$ antiferromagnetic easy-axis XXZ model on a square and a honeycomb lattice. Describing the excitations and interactions of magnons in these models is of substantial interest, as magnons are the fundamental quasiparticles and can thus explain experimental observations. Already for the paradigmatic antiferromagnetic spin-$\frac{1}{2}$ Heisenberg model on the square lattice, high-energy features of the magnon dispersion are understood via strong magnon-magnon interactions. Besides that, interesting high-energy features were recently observed in the honeycomb lattice, including a possible decay of the single-magnon mode. The goal of this thesis is to obtain effective descriptions of the aforementioned systems that capture the relevant features of the magnon excitations and their interactions. To study these features, this thesis employs continuous similarity transformations (CSTs), which enables a systematic derivation of an effective quasiparticle picture via so-called flow equations. The approach enables decoupling different magnon sectors and thus for investigating ground-state properties, single-magnon excitations, and multi-magnon bound states. The magnon description is derived from the non-Hermitian Dyson-Maleev representation, which describes fluctuations around a long-range magnetically ordered ground state. The flow equations are set up in momentum space, truncated by the scaling dimension, retaining only operators up to scaling dimension of two, and solved numerically. For the square lattice, a magnon-conserving effective Hamiltonian is obtained across the full anisotropy range by interpolating between the Ising limit and the Heisenberg limit. The ground-state energy, staggered magnetization, dispersion, and critical exponents are in excellent agreement with the literature, supporting the validity of CST also for gapped phases. Two-magnon excitations in the $S^z=0$ subspace comprise four bound states, which are tracked across the anisotropy range. Their successive decay into the two-magnon continuum is determined by using the inverse participation ratio. For the honeycomb lattice, the magnon-conserving CST is reliable only up to $\lambda ≲ 0.57$; an adapted CST scheme that decouples only the ground state while preserving couplings between different higher-magnon sectors yields effective descriptions for all \(\lambda\) and confirms stable long-range order. The low-energy properties determined in this way match the literature values. However, in the Heisenberg limit, the adapted approach fails to capture high-energy features, such as the experimentally observed decay into the multi-particle continuum at corners of the Brillouin zone. A termination of the magnon-conserving flow equations at a finite flow parameter before the divergence reveals a single-magnon decay at high energies, which is connected to a crossing of a two-magnon bound state with the single-magnon mode. The observed energy dip qualitatively agrees with Quantum Monte Carlo results, but magnon–magnon interactions are overestimated, hindering a fully quantitative characterization of the single-magnon dispersion. Analyzing multi-magnon bound states over the anisotropy range unveils that the crossing of a three-magnon bound state with the single-magnon mode accounts for the breakdown of the magnon-conserving CST at higher $\lambda$. This work demonstrates that CST with scaling-dimension truncation provides a powerful framework for studying two-dimensional quantum magnets, capturing low-energy properties and bound states. Nevertheless, challenges remain in accurately describing high-energy features arising from a persistent overlap of magnon sectors due to strong binding effects.

•Jan 1, 2026•0 citations

Study of the excited states of the quantum antiferromagnets

Mohammad G. Merdan

We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic \emph{XXZ} model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone $\Gamma(\mathbf q=0)$ and at the magnetic ordering wavevectors $\pm[\mathbf Q= (4\pi/3,0)]$, the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.

•Nov 10, 2013•0 citations

Spinons, solitons, and magnons in one-dimensional Heisenberg-Ising antiferromagnets

Rajiv R. P. Singh

We calculate the excitation spectra for the 1D Heisenberg-Ising antiferromagnets by expansions around the Ising limit. For S=1/2, the calculated expansion coefficients for the spinon spectra agree term by term with the solution of Johnson and McCoy. For S=1, the solitons become gapless before the Heisenberg limit is reached, signaling a transition to the Haldane phase. By applying a staggered field we calculate the one-magnon spectra for the S=1 Heisenberg chain. For S=3/2 the quantum renormalization of the spin-wave spectra is calculated to be approximately 1.16. \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9510064•Physical review. B, Condensed matter•May 1, 1996•2 citations
We employ spin-cluster perturbation theory to investigate the evolution of the spin excitation spectra of the mixed-dimensional antiferromagnetic s=12s=\frac{1}{2}s=21​ Heisenberg models from the one-dimensional chain to the two-dimensional square and honeycomb lattices by varying the spin interactions J\ensuremath⊥{J}_{\ensuremath{\perp}}J\ensuremath⊥​ across the chains relative to J\ensuremath∥{J}_{\ensuremath{\parallel}}J\ensuremath∥​ along the chains. In addition to the well-defined magnon excitations, the spectra of the N'eel phases for both lattices exhibit anomalous high-energy continua that evolve continuously from the continuum of one-dimensional chain through increasing J\ensuremath⊥{J}_{\ensuremath{\perp}}J\ensuremath⊥​. Notably, the key features of the continua at the two-dimensional limit for J\ensuremath⊥/J\ensuremath∥=1{J}_{\ensuremath{\perp}}/{J}_{\ensuremath{\parallel}}=1J\ensuremath⊥​/J\ensuremath∥​=1 are well consistent with experimental observations in real materials. Our results indicate that these high-energy continua in the N'eel phases are a result of spinon deconfinement, which is further supported by a random-phase-approximation analysis based on the spinon Hubbard models.
We employ the spin cluster perturbation theory to investigate the dynamical properties of the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the honeycomb lattice. We obtain the excitation spectra for all possible phases in the phase diagram, including the N\'eel phase, plaquette valence-bond-solid phase, dimer valence-bond-solid phase, and stripe antiferromagnetic phase. In the N\'eel phase, besides the obvious renormalization of the magnon dispersion, we find that the spectrum exhibits a dome-shaped broad continuum around the second Brillouin zone (BZ) and the additional strong continuum close to the corner of the BZ. In the valence-bond-solid phases, the spectra are dominated by a strong broad continuum all the way down to below J1{J}_{1}J1​ coexisting with the lowest-energy triplon modes characterizing the plaquette and dimer phases. We ascribe this strong broad continuum and the additional continuum close to the BZ corner in the N\'eel phase to the contributions of fractionalized spinon excitations. In the stripe phase, a clear difference from the linear-spin-wave approximation is that the spectrum is gapped at the MMM point, while that obtained by the latter is gapless due to the strong quantum fluctuations. We point out that the features observed in the N\'eel phase are consistent with the recent neutron scattering experiments on YbCl3{\mathrm{YbCl}}_{3}YbCl3​ and YbBr3{\mathrm{YbBr}}_{3}YbBr3​.
Spin excitation spectra of two dimensional S=1/2S=1/2S=1/2 Heisenberg model with checkerboard structures.
We study the excitation spectra of two-dimensional spin-1/21/21/2 Heisenberg model with checkerboard structures using stochastic analytic continuation of imaginary-time correlation function obtained from quantum Monte Carlo simulation. The checkerboard models have two different antiferromagnetic nearest-neighbor interactions J1J_{1}J1​ and J2J_{2}J2​, and the tuning parameter ggg is defined as J2/J1J_{2}/J_{1}J2​/J1​. The dynamic structure factors are systematically calculated in all phases of the models as well as at the critical points. To give a full understanding of the dynamic spectra, wave theory is employed to explain some features of numerical results, especially for the low-energy part. When ggg is close to 111, the features of the excitation spectra of each checkerboard model are roughly the same as those of the original square lattice antiferromagnetic Heisenberg model, and the high-energy continuum among them are discussed. In contrast to the other checkerboard structures investigated in this paper, the 3×33\times 33×3 checkerboard model has distinctive excitation features, like a gap between a low-energy gapless branch and a gapped high-energy part exists when ggg is small. The gapless branch in this case can be regarded as wave in Neˊ\mathrm{\acute{e}}eˊel order formed by block spin in each 3×33\times 33×3 plaquette with effective exchange interaction originated from renormalization. One finding is that the continuum also appears in this low-energy branch, which may originated from nearly deconfined process in block spin.
We study the spin excitation spectra of the two-dimensional spin-12\frac{1}{2}21​ Heisenberg model with a checkerboard structure using stochastic analytic continuation of the imaginary-time correlation function obtained from a quantum Monte Carlo simulation. The checkerboard models have two different antiferromagnetic nearest-neighbor interactions J1{J}_{1}J1​ and J2{J}_{2}J2​, and the tuning parameter ggg is defined as J2/J1{J}_{2}/{J}_{1}J2​/J1​. The dynamic spin structure factors are systematically calculated in all phases of the models as well as at the critical points. To give a full understanding of the dynamic spectra, spin wave theory is employed to explain some features of numerical results, especially for the low-energy part. When ggg is close to 1, the features of the spin excitation spectra of each checkerboard model are roughly the same as those of the original square lattice antiferromagnetic Heisenberg model, and the high-energy continuum among them is discussed. In contrast to the other checkerboard structures investigated in this paper, the $3\ifmmode\times\else\texttimes\fi{}3$ checkerboard model has distinctive excitation features, such as a gap between a low-energy gapless branch and a gapped high-energy part that exists when ggg is small. The gapless branch in this case can be regarded as a spin wave in N\'eel order formed by a ``block spin'' in each $3\ifmmode\times\else\texttimes\fi{}3$ plaquette with an effective exchange interaction originating from renormalization. One unexpected finding is that the continuum also appears in this low-energy branch.
This thesis investigates two-dimensional quantum antiferromagnets, with a focus on the spin-12\frac{1}{2}21​ antiferromagnetic easy-axis XXZ model on a square and a honeycomb lattice. Describing the excitations and interactions of magnons in these models is of substantial interest, as magnons are the fundamental quasiparticles and can thus explain experimental observations. Already for the paradigmatic antiferromagnetic spin-12\frac{1}{2}21​ Heisenberg model on the square lattice, high-energy features of the magnon dispersion are understood via strong magnon-magnon interactions. Besides that, interesting high-energy features were recently observed in the honeycomb lattice, including a possible decay of the single-magnon mode. The goal of this thesis is to obtain effective descriptions of the aforementioned systems that capture the relevant features of the magnon excitations and their interactions. To study these features, this thesis employs continuous similarity transformations (CSTs), which enables a systematic derivation of an effective quasiparticle picture via so-called flow equations. The approach enables decoupling different magnon sectors and thus for investigating ground-state properties, single-magnon excitations, and multi-magnon bound states. The magnon description is derived from the non-Hermitian Dyson-Maleev representation, which describes fluctuations around a long-range magnetically ordered ground state. The flow equations are set up in momentum space, truncated by the scaling dimension, retaining only operators up to scaling dimension of two, and solved numerically. For the square lattice, a magnon-conserving effective Hamiltonian is obtained across the full anisotropy range by interpolating between the Ising limit and the Heisenberg limit. The ground-state energy, staggered magnetization, dispersion, and critical exponents are in excellent agreement with the literature, supporting the validity of CST also for gapped phases. Two-magnon excitations in the Sz=0S^z=0Sz=0 subspace comprise four bound states, which are tracked across the anisotropy range. Their successive decay into the two-magnon continuum is determined by using the inverse participation ratio. For the honeycomb lattice, the magnon-conserving CST is reliable only up to λ≲0.57\lambda ≲ 0.57λ≲0.57; an adapted CST scheme that decouples only the ground state while preserving couplings between different higher-magnon sectors yields effective descriptions for all λ\lambdaλ and confirms stable long-range order. The low-energy properties determined in this way match the literature values. However, in the Heisenberg limit, the adapted approach fails to capture high-energy features, such as the experimentally observed decay into the multi-particle continuum at corners of the Brillouin zone. A termination of the magnon-conserving flow equations at a finite flow parameter before the divergence reveals a single-magnon decay at high energies, which is connected to a crossing of a two-magnon bound state with the single-magnon mode. The observed energy dip qualitatively agrees with Quantum Monte Carlo results, but magnon–magnon interactions are overestimated, hindering a fully quantitative characterization of the single-magnon dispersion. Analyzing multi-magnon bound states over the anisotropy range unveils that the crossing of a three-magnon bound state with the single-magnon mode accounts for the breakdown of the magnon-conserving CST at higher λ\lambdaλ. This work demonstrates that CST with scaling-dimension truncation provides a powerful framework for studying two-dimensional quantum magnets, capturing low-energy properties and bound states. Nevertheless, challenges remain in accurately describing high-energy features arising from a persistent overlap of magnon sectors due to strong binding effects.
We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic \emph{XXZ} model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone Γ(q=0)\Gamma(\mathbf q=0)Γ(q=0) and at the magnetic ordering wavevectors ±[Q=(4π/3,0)]\pm[\mathbf Q= (4\pi/3,0)]±[Q=(4π/3,0)], the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.
We calculate the excitation spectra for the 1D Heisenberg-Ising antiferromagnets by expansions around the Ising limit. For S=1/2, the calculated expansion coefficients for the spinon spectra agree term by term with the solution of Johnson and McCoy. For S=1, the solitons become gapless before the Heisenberg limit is reached, signaling a transition to the Haldane phase. By applying a staggered field we calculate the one-magnon spectra for the S=1 Heisenberg chain. For S=3/2 the quantum renormalization of the spin-wave spectra is calculated to be approximately 1.16. \textcopyright{} 1996 The American Physical Society.