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/0503135 · Physical Review B

S=12 chain in a staggered field: High-energy bound-spinon state and the effects of a discrete lattice

M. Kenzelmann, Cristian D. Batista, Y. Chen, C. Broholm, Daniel H. Reich, S. Park, Yunfeng Qiu

Published Mar 11, 2005• 22 citations
arXivHTMLPDF

Abstract

We report an experimental and theoretical study of the antiferromagnetic $S=\frac{1}{2}$ chain subject to uniform and staggered fields. Using inelastic neutron scattering, we observe a bound-spinon state at high energies in the linear chain compound ${\mathrm{CuCl}}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}2({({\mathrm{CD}}_{3})}_{2}\mathrm{SO})$. The excitation is explained with a mean-field theory of interacting $S=\frac{1}{2}$ fermions and arises from the opening of a gap at the Fermi surface due to confining spinon interactions. The mean-field model also describes the wave-vector dependence of the bound-spinon states, particularly in regions where effects of the discrete lattice are important. We calculate the dynamic structure factor using exact diagonalization of finite length chains, obtaining excellent agreement with the experiments.

DOI
10.1103/physrevb.71.094411

Related papers

Bound Spinons in an Antiferromagnetic S=1/2 Chain with a Staggered Field

M. Kenzelmann, Yinghao Chen, C. Broholm +2 more

Inelastic neutron scattering was used to measure the magnetic field dependence of spin excitations in the antiferromagnetic $S=1/2$ chain ${\mathrm{C}\mathrm{u}\mathrm{C}\mathrm{l}}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}2(\mathrm{\text{dimethylsulfoxide}})$ in the presence of uniform and staggered fields. Dispersive bound states emerge from a zero-field two-spinon continuum with different finite energy minima at wave numbers $q=\ensuremath{\pi}$ and ${q}_{i}\ensuremath{\approx}\ensuremath{\pi}(1\ensuremath{-}2\ensuremath{\langle}{S}_{z}\ensuremath{\rangle})$. The ratios of the field dependent excitation energies are in excellent agreement with predictions for breather and soliton solutions to the quantum sine-Gordon model, the proposed low-energy theory for $S=1/2$ chains in a staggered field. The data are also consistent with the predicted soliton and $n=1,2$ breather polarizations and scattering cross sections.

arXiv:cond-mat/0305476•Physical Review Letters•Jul 2, 2004•71 citations

Spinon confinement in the gapped antiferromagnetic XXZ spin- 12 chain

S. B. Rutkevich

The infinite Heisenberg XXZ spin-$\frac{1}{2}$ chain in the gapped antiferromagnetic regime has two degenerate vacua and kink topological excitations (which are also called spinons) interpolating between these vacua as elementary excitations. Application of an arbitrary weak staggered longitudinal magnetic field $h$ induces a long-range attractive potential between two adjacent spinons leading to their confinement into ``meson'' bound states. Exploiting the integrability of the XXZ model in the deconfined phase $h=0$, we perform perturbative calculations of the energy spectra of the two-spinon bound states in the weak confinement regime at $h\ensuremath{\rightarrow}+0$, using the strength of the staggered magnetic field $h$ as a small parameter. Both transverse and longitudinal dynamical structure factors of the local spin operators are calculated as well in the two-spinon approximation in the weak confinement regime to the leading order in $h$.

arXiv:2207.12215•Physical review. B./Physical review. B•Oct 7, 2022•8 citations

Confinement of spinons in the XXZ spin- 12 chain in presence of a transverse magnetic field

S. B. Rutkevich

We study the tuning effect of a transverse magnetic field on the confinement of spinons in the infinite XXZ spin-1/2 chain. The spinon confinement in this model takes place in the gapped antiferromagnetic phase upon application of a staggered longitudinal magnetic field. The tuning transverse magnetic field has mutually orthogonal uniform and staggered components. The energy spectra of the two-spinon bound states (the ``mesons'') in the confinement regime are analytically calculated in this model using two different perturbative schemes. The first one applies in the extreme anisotropic (Ising) limit and employs the inverse anisotropy constant as a small parameter. The second perturbative scheme, which applies at any anisotropy in the gapped antiferromagnetic domain, exploits the integrability of the XXZ spin chain at zero magnetic field. The small parameters in the second technique are the components of the transverse, and staggered longitudinal magnetic fields. It is shown that the weak transverse magnetic field mixes the transverse and longitudinal meson modes, and leads to an avoided crossing of their energies upon increase of its strength. The explicit formulas for the two-spinon contribution to the dynamical structure factors of local spin operators are obtained as well in this model in the weak confinement regime for wave vectors close to the points $k=0$ and $\ensuremath{\pi}$.

arXiv:2307.08328•Physical review. B./Physical review. B•Jan 12, 2024•2 citations

Field-induced gap in Cu benzoate and other S=12 antiferromagnetic chains

Ian K. Affleck, Masaki Oshikawa

Recent experiments on the $S=\frac{1}{2}$ antiferromagnetic chain compound, Cu benzoate, discovered an unexpected gap scaling as approximately the $\frac{2}{3}$ power of an applied magnetic field. A theory of this gap, based on an effective staggered field, orthogonal to the applied uniform field, resulting from a staggered gyromagnetic tensor and a Dzyaloshinskii-Moriya interaction, leading to a sine-Gordon quantum field theory, has been developed. Here we discuss many aspects of this subject in considerable detail, including a review of the $S=\frac{1}{2}$ chain in a uniform field, a spin-wave theory analysis of the uniform plus staggered field problem, exact amplitudes for the scaling of gap, staggered susceptibility and staggered magnetization with field or temperature, intensities of soliton and breather peaks in the structure function, and field and temperature dependence of the total susceptibility.

arXiv:cond-mat/9905002•Physical review. B, Condensed matter•Jul 1, 1999•309 citations

Neutron Scattering Studies of Excitations in Spin Chains

E. Hirtenlechner

Many phenomena in magnetism can be described using classical models, yet magnetism is a quantum mechanical effect. In addition to thermal fluctuations, quantum fluctuations arise due to the quantum uncertainty of the ground state. These fluctuations are enhanced by low dimensionality and low spin. Spin chains show interesting excitations associated with quasiparticles. Model materials for spin chains can be realized in compounds where the interchain interactions are weak enough. This thesis presents neutron scattering studies of two model materials for spin chains with different spin symmetry. The spin2 Heisenberg antiferromagnetic chain CuSO4*5D2O has a main exchange of J ∼ 3 K, which permits to study its excitation spectrum up to temperatures of ten times the exchange. The spin2 Heisenberg magnet has been studied widely both theoretically and experimentally. The excitation spectrum of the spin2 Heisenberg antiferromagnet consists of quasiparticles called spinons, and calculations for it exist for T = 0 and as a function of temperature. A quantitative comparison between experimental data and theoretical models across a range of temperatures is presented. The excitation spectrum of CuSO4*5D2O is excellently described by the excitation spectrum of a spin2 Heisenberg antiferromagnetic chain at finite temperatures. Energy-temperature scaling is observed at temperatures up to 2J . For these temperatures the Luttinger liquid description of the spin2 Heisenberg antiferromagnetic chain is valid. A spinon continuum is confirmed at all measured temperatures. Therefore, even at ten times the main exchange (T ≈ 10 J), CuSO4*5D2O shows the behavior of a quantum magnet, and not of a classical system at infinite temperatures, for which one would expect single spin flips. The Ising chain is one of the most studied models in magnetism, one model compound for it is RbCoCl3. Optical spectroscopy measurements suggest a lower gap than for similar compounds, which makes it better suited for high resolution neutron spectroscopy. Magnetic correlations along the chains arise in RbCoCl3 below 80 K. A gapped continuum is observed, which arises from isolated antiferromagnetic Ising chains. Below the first magnetic ordering temperature TN1 ≈ 28 K bound states are observed, which arise due to correlations which form between the chains. Strong diffuse scattering leads to the conclusion that disorder is still present. At the second magnetic ordering transition, at TN2 ≈ 14 K, the bound states split as the correlations between the chains change. Fits indicate that the correlations change continuously as a function of temperature. Above TN2 the excitation spectrum of RbCoCl3 can be described by a spin1 2 Ising chain with an XY-component which is treated as a perturbation. Below TN2 the main features are reproduced correctly, but deviations from the details indicate that additional terms in the Hamiltonian may be necessary. This thesis was carried out at the Institut Laue-Langevin, the Laboratory for Neutron Scattering and Imaging at the Paul Scherrer Institute and ETH Zurich.

•Jan 1, 2014•0 citations

Effect of Staggered Field on the Magnetization Process ofS= 1/2 Antiferromagnetic Chains

Akira Matsuo, Kohei Hosokawa, Yasuo Narumi +7 more

We measured the magnetization processes of the S = 1/2 antiferromagnetic chain compound, Cu benzoate. From the inelastic neutron scattering and the specific heat measurements it was recently discovered that the substance exhibits an unexpected gap which varies approximately as the 2/3 power of an applied magnetic field. The primary mechanism of the gap formation is produced by an effective staggered field due to both the alternating g-tensor and the Dzyaloshinskii-Moriya interaction. We observed the effect of the effective staggered field on the magnetization process of Cu benzoate as a rounding behavior near the saturation field as well as an initial nonlinear increment near zero field. The experimental results are well reproduced by the numerical calculation on the Hamiltonian which includes the appropriate term of the staggered field, using the finite temperature density matrix renormalization group method.

Progress of Theoretical Physics Supplement•Jan 1, 2005•3 citations
We report an experimental and theoretical study of the antiferromagnetic S=12S=\frac{1}{2}S=21​ chain subject to uniform and staggered fields. Using inelastic neutron scattering, we observe a bound-spinon state at high energies in the linear chain compound CuCl2\ifmmode⋅\else\textperiodcentered\fi2((CD3)2SO){\mathrm{CuCl}}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}2({({\mathrm{CD}}_{3})}_{2}\mathrm{SO})CuCl2​\ifmmode⋅\else\textperiodcentered\fi2((CD3​)2​SO). The excitation is explained with a mean-field theory of interacting S=12S=\frac{1}{2}S=21​ fermions and arises from the opening of a gap at the Fermi surface due to confining spinon interactions. The mean-field model also describes the wave-vector dependence of the bound-spinon states, particularly in regions where effects of the discrete lattice are important. We calculate the dynamic structure factor using exact diagonalization of finite length chains, obtaining excellent agreement with the experiments.
Inelastic neutron scattering was used to measure the magnetic field dependence of spin excitations in the antiferromagnetic S=1/2S=1/2S=1/2 chain ${\mathrm{C}\mathrm{u}\mathrm{C}\mathrm{l}}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}2(\mathrm{\text{dimethylsulfoxide}})$ in the presence of uniform and staggered fields. Dispersive bound states emerge from a zero-field two-spinon continuum with different finite energy minima at wave numbers $q=\ensuremath{\pi}$ and ${q}_{i}\ensuremath{\approx}\ensuremath{\pi}(1\ensuremath{-}2\ensuremath{\langle}{S}_{z}\ensuremath{\rangle})$. The ratios of the field dependent excitation energies are in excellent agreement with predictions for breather and soliton solutions to the quantum sine-Gordon model, the proposed low-energy theory for S=1/2S=1/2S=1/2 chains in a staggered field. The data are also consistent with the predicted soliton and n=1,2n=1,2n=1,2 breather polarizations and scattering cross sections.
The infinite Heisenberg XXZ spin-12\frac{1}{2}21​ chain in the gapped antiferromagnetic regime has two degenerate vacua and kink topological excitations (which are also called spinons) interpolating between these vacua as elementary excitations. Application of an arbitrary weak staggered longitudinal magnetic field hhh induces a long-range attractive potential between two adjacent spinons leading to their confinement into ``meson'' bound states. Exploiting the integrability of the XXZ model in the deconfined phase h=0h=0h=0, we perform perturbative calculations of the energy spectra of the two-spinon bound states in the weak confinement regime at $h\ensuremath{\rightarrow}+0$, using the strength of the staggered magnetic field hhh as a small parameter. Both transverse and longitudinal dynamical structure factors of the local spin operators are calculated as well in the two-spinon approximation in the weak confinement regime to the leading order in hhh.
We study the tuning effect of a transverse magnetic field on the confinement of spinons in the infinite XXZ spin-1/2 chain. The spinon confinement in this model takes place in the gapped antiferromagnetic phase upon application of a staggered longitudinal magnetic field. The tuning transverse magnetic field has mutually orthogonal uniform and staggered components. The energy spectra of the two-spinon bound states (the ``mesons'') in the confinement regime are analytically calculated in this model using two different perturbative schemes. The first one applies in the extreme anisotropic (Ising) limit and employs the inverse anisotropy constant as a small parameter. The second perturbative scheme, which applies at any anisotropy in the gapped antiferromagnetic domain, exploits the integrability of the XXZ spin chain at zero magnetic field. The small parameters in the second technique are the components of the transverse, and staggered longitudinal magnetic fields. It is shown that the weak transverse magnetic field mixes the transverse and longitudinal meson modes, and leads to an avoided crossing of their energies upon increase of its strength. The explicit formulas for the two-spinon contribution to the dynamical structure factors of local spin operators are obtained as well in this model in the weak confinement regime for wave vectors close to the points k=0k=0k=0 and $\ensuremath{\pi}$.
Recent experiments on the S=12S=\frac{1}{2}S=21​ antiferromagnetic chain compound, Cu benzoate, discovered an unexpected gap scaling as approximately the 23\frac{2}{3}32​ power of an applied magnetic field. A theory of this gap, based on an effective staggered field, orthogonal to the applied uniform field, resulting from a staggered gyromagnetic tensor and a Dzyaloshinskii-Moriya interaction, leading to a sine-Gordon quantum field theory, has been developed. Here we discuss many aspects of this subject in considerable detail, including a review of the S=12S=\frac{1}{2}S=21​ chain in a uniform field, a spin-wave theory analysis of the uniform plus staggered field problem, exact amplitudes for the scaling of gap, staggered susceptibility and staggered magnetization with field or temperature, intensities of soliton and breather peaks in the structure function, and field and temperature dependence of the total susceptibility.
Many phenomena in magnetism can be described using classical models, yet magnetism is a quantum mechanical effect. In addition to thermal fluctuations, quantum fluctuations arise due to the quantum uncertainty of the ground state. These fluctuations are enhanced by low dimensionality and low spin. Spin chains show interesting excitations associated with quasiparticles. Model materials for spin chains can be realized in compounds where the interchain interactions are weak enough. This thesis presents neutron scattering studies of two model materials for spin chains with different spin symmetry. The spin2 Heisenberg antiferromagnetic chain CuSO4*5D2O has a main exchange of J ∼ 3 K, which permits to study its excitation spectrum up to temperatures of ten times the exchange. The spin2 Heisenberg magnet has been studied widely both theoretically and experimentally. The excitation spectrum of the spin2 Heisenberg antiferromagnet consists of quasiparticles called spinons, and calculations for it exist for T = 0 and as a function of temperature. A quantitative comparison between experimental data and theoretical models across a range of temperatures is presented. The excitation spectrum of CuSO4*5D2O is excellently described by the excitation spectrum of a spin2 Heisenberg antiferromagnetic chain at finite temperatures. Energy-temperature scaling is observed at temperatures up to 2J . For these temperatures the Luttinger liquid description of the spin2 Heisenberg antiferromagnetic chain is valid. A spinon continuum is confirmed at all measured temperatures. Therefore, even at ten times the main exchange (T ≈ 10 J), CuSO4*5D2O shows the behavior of a quantum magnet, and not of a classical system at infinite temperatures, for which one would expect single spin flips. The Ising chain is one of the most studied models in magnetism, one model compound for it is RbCoCl3. Optical spectroscopy measurements suggest a lower gap than for similar compounds, which makes it better suited for high resolution neutron spectroscopy. Magnetic correlations along the chains arise in RbCoCl3 below 80 K. A gapped continuum is observed, which arises from isolated antiferromagnetic Ising chains. Below the first magnetic ordering temperature TN1 ≈ 28 K bound states are observed, which arise due to correlations which form between the chains. Strong diffuse scattering leads to the conclusion that disorder is still present. At the second magnetic ordering transition, at TN2 ≈ 14 K, the bound states split as the correlations between the chains change. Fits indicate that the correlations change continuously as a function of temperature. Above TN2 the excitation spectrum of RbCoCl3 can be described by a spin1 2 Ising chain with an XY-component which is treated as a perturbation. Below TN2 the main features are reproduced correctly, but deviations from the details indicate that additional terms in the Hamiltonian may be necessary. This thesis was carried out at the Institut Laue-Langevin, the Laboratory for Neutron Scattering and Imaging at the Paul Scherrer Institute and ETH Zurich.
We measured the magnetization processes of the S = 1/2 antiferromagnetic chain compound, Cu benzoate. From the inelastic neutron scattering and the specific heat measurements it was recently discovered that the substance exhibits an unexpected gap which varies approximately as the 2/3 power of an applied magnetic field. The primary mechanism of the gap formation is produced by an effective staggered field due to both the alternating g-tensor and the Dzyaloshinskii-Moriya interaction. We observed the effect of the effective staggered field on the magnetization process of Cu benzoate as a rounding behavior near the saturation field as well as an initial nonlinear increment near zero field. The experimental results are well reproduced by the numerical calculation on the Hamiltonian which includes the appropriate term of the staggered field, using the finite temperature density matrix renormalization group method.