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

Journal of Physics Condensed Matter

Two-spin-wave spectra of easy-plane ferromagnetic chains

Peter S. Riseborough, Pradeep Kumar

Published Oct 9, 1989• 3 citations
Publisher

Abstract

The easy-plane ferromagnetic chain in an applied magnetic field is known to map onto a sine-Gordon field theory, in the classical continuum limit. The sine-Gordon field theory supports breather excitations which, when quantised, are identifiable as a hierarchy of bound states of several small-amplitude excitations. For increasing values of the interaction parameter, the highest members of the hierarchy become unstable to the emission of a soliton/anti-soliton pair. For a critical value of the interaction parameter, even the ground state of the quantised sine-Gordon theory becomes unstable. For the quantum spin system, the excitations corresponding to the two lowest members of the hierarchy of breathers are the single-spin-wave and the two-spin-wave bound states. The authors examine the two-spin-wave bound-state excitations of the quantum spin system directly, using a large-S approach. Although the method does rely on large S, they recover previously derived results valid for arbitrary S, such as the exact two-spin-wave bound states for the isotropic Heisenberg ferromagnet and the expansion in the anisotropy about this result. In the limit of large anisotropy the two-spin-wave bound state becomes unstable, in a manner suggestive of the instability of the quantum-sine-Gordon field theory.

DOI
10.1088/0953-8984/1/40/018

Related papers

Classical aspects of quantum spin chains (abstract)

L. F. Lemmens, Wim J. M. de Jonge, Klaas Kopinga

A variational estimate of the free energy of a quantum spin chain based on the stochastic behavior of the quantization axes is proposed. It allows a description of the nonlinear excitations and quantum spin waves within the same framework. For an easy-plane ferromagnet with an external field in the plane it reveals that the nonlinear excitations (solitons) are predominantly described by the classical degrees of freedom. Considerable effort has been devoted to the theoretical description1 and experimental verification of the existence of nonlinear excitations. In this field magnetic chain systems have played an important part.2 Chain systems with an easy-plane anisotropy, such as (C6 H11 NH3 )CuBr3 (CHAB) and CsNiF3 , which are subjected to a symmetry breaking field, were found to be soliton bearing.3 Theoretically the description of the nonlinear effect is based on the mapping of the spin dynamics on a sine-Gordon model. Further research4 indicated that the interpretation of the thermodynamical properties of real systems in terms of the sine-Gordon model required however quantum and anisotropy corrections. Since the dipolar interactions between the spins in the chain are considered to be negligible, the direction of the spin chain in configuration space with respect to any preferred direction in spin space does not enter explicitly in the Hamiltonian. This makes it interesting to define a site-dependent z direction in spin space which makes an angle (Θn ,φn ) with the z direction in configuration space. The model Hamiltonian can then be written in terms of the spin operators defined with respect to the site-dependent quantization axis. The representation with equal quantization axis for all the sites is related to the representation with different quantization axes at each site by a rotation in spin space. The unitary transformation that describes the rotation contains the angles (Θn ,φn ), which we have treated as random variables. Based on this observation we have proposed a new variational approach for the quantum spin chain. We will outline the main results here. A full account is published elsewhere.5 In our approach the free energy of the system can be separated in a quantum part and a classical part. The free energy F of the chain is then approximated by an upper bound F<Fcl +Fqm , where βFcl =−ln[E(exp−βVcl )] and βFqm =−ln Tr(exp−β〈Hqm 〉). The first part is a configurational classical free energy, depending on the choice of the probability distribution for the quantization axes via the expectation E. The second contribution is the free energy of a modified quantum chain, whose parameters are classical averages and can be obtained from the configurational classical free energy. For easy-plane ferromagnetic chains, the nonlinear excitations are comprised in the classical part, whereas the quantum part contains predominantly linear excitations. For those thermodynamical quantities that are insensitive to the contribution of the quantum excitations, the chain is thermodynamically equivalent with a classical sine-Gordon system. The out-of-plane corrections are incorporated in the quantum part of the free energy. This behavior is corroborated experimentally by measurements of the excess heat capacity of CHAB for different magnetic fields with the same in-plane component.6 The validity of our approach is, however, not restricted to easy-plane ferromagnetic chains. Also for more complex systems with, for example, an orthorhombic or helical symmetry, the introduction of stochastic quantization axes can shed new light on some of the thermodynamical properties of these quantum systems in terms of their classical equivalent.

Journal of Applied Physics•May 1, 1990•0 citations

Spin Dynamics in Quantum Sine-Gordon Spin Chains: High-Field ESR Studies

Sergei A. Zvyagin

Abstract A spin-1/2 Heisenberg antiferromagnetic chain is one of the most important paradigmatic models in quantum magnetism. Its ground state is a spin singlet, while the excitation spectrum is formed by gapless fractional excitations, spinons. The presence of alternating g-tensors and/or the staggered Dzyaloshinskii-Moriya interaction results in opening the energy gap $$\varDelta \propto H^{2/3}$$ Δ ∝ H 2 / 3 , once the magnetic field H is applied. A fairly good understanding of this phenomenon was achieved in the framework of the sine-Gordon quantum-field theory, taking into account the effective transverse staggered field induced by the applied uniform field. The theory predicts solitons and antisolitons as elementary excitations, as well as their bound states, breathers. Here, I review recent high-field electron spin resonance spectroscopy studies of such systems, focusing on peculiarities of their spin dynamics in the sine-Gordon regime and beyond.

Applied Magnetic Resonance•Feb 24, 2021•5 citations

Bound states and correlation functions in the easy-plane Heisenberg ferromagnetic chain

H J Jensen, Hans C. Fogedby

The two-magnon excitation spectrum of the quantum spin-one easy-plane Heisenberg ferromagnetic chain is discussed. The spin-spin correlation functions are calculated within a systematic perturbation theory in terms of the anisotropy, and contributions from the two-magnon bound states are discussed. The results are compared with the predictions for non-interacting spin waves.

Journal of Physics C Solid State Physics•Feb 28, 1987•0 citations

Soliton energy in an easy-plane quantum spin chain

H.-J. Mikeska

We investigate the quantum corrections to the energy of the static soliton in the $\mathrm{XY}$-like ferromagnetic spin chain in a symmetry-breaking external field. We apply a semiclassical continuum approximation which allows us to treat the leading-order deviations from idealized planar behavior. The equivalent quantum-field theory, an extension of the quantum sine-Gordon theory, is renormalized and its soliton energy is shown to depend strongly on this deviation. The quantum-field-theoretical renormalization, however, does not affect the energy of the magnetic soliton, which is calculated to leading order in $\frac{1}{S}$, where $S$ is the spin length. The sine-Gordon limit is found applicable only for very large anisotropies; for realistic anisotropies, the reduction of the soliton energy owing to quantum effects is enhanced and depends only on $S$. A quantitative calculation for the $S=1$ system CsNi${\mathrm{F}}_{3}$ is likely to require higher-order terms in $\frac{1}{S}$.

Physical review. B, Condensed matter•Nov 1, 1982•74 citations

Spinon, soliton and breather in the spin-1/2 antiferromagnetic chain KCuGaF$_6$

Izumi Umegaki, Hidekazu Tanaka, Nobuyuki Kurita +7 more

Elementary excitations of the S=1/2 one-dimensional antiferromagnet KCuGaF_6 were investigated by inelastic neutron scattering in zero and finite magnetic fields perpendicular to the (1, 1, 0) plane combined with specific heat measurements. KCuGaF$_6$ exhibits no long-range magnetic ordering down to 50 mK despite the large exchange interaction J/k_B=103 K. At zero magnetic field, well-defined spinon excitations were observed. The energy of the des Cloizeaux and Pearson mode of the spinon excitations is somewhat larger than that calculated with the above exchange constant. This discrepancy is mostly ascribed to the effective XY anisotropy arising from the large Dzyaloshinsky-Moriya interaction with an alternating D vector. KCuGaF_6 in a magnetic field is represented by the quantum sine-Gordon model, for which low-energy elementary excitations are composed of solitons, antisolitons and their bound states called breathers. Unlike the theoretical prediction, it was found that the energy of a soliton is smaller than that of the first breather, although the energy of the first breather coincides with that observed in a previous ESR measurement.

arXiv:1510.08200•Oct 28, 2015•0 citations

Solitons and magnon bound states in ferromagnetic Heisenberg chains

T. Schneider

We investigate the relation between magnon, magnon bound states, and the classical soliton solutions in the isotropic, the anisotropic-exchange, and the easy-axis ferromagnetic Heisenberg chains. The Dyson-Maleev boson representation is used to study the interaction between magnons, and bound states are investigated in terms of two-spin Green's functions. For easy-axis and anisotropic-exchange magnets, a mapping to the Bose gas with attractive $\ensuremath{\delta}$-function interactions is established, and it yields the eigenvalues of the magnon and the $m$th bound state in the weak-coupling and continuum limits. The classical limit of the two-spin Green's function reveals that, to leading order in temperature, the bound-state resonance and the associated effects on the two-magnon continuum survives. An important result is a ${T}^{2}$ dependence of the "binding" energy of the bound-state resonance. As a consequence, in a classical description the bound states enter only in order ${T}^{2}$. Finally, we quantize the soliton solutions according to the Bohr-Sommerfeld and de Broglie rules. This approach is found to be exact for the $s=\frac{1}{2}$ isotropic Heisenberg chain and for sufficiently small wave numbers for all $s$ values. In the anisotropic-exchange and easy-axis models, it agrees with the results obtained from the mapping on the Bose gas, for large quantum numbers and in the easy-axis case for $s\ensuremath{\gg}\frac{1}{2}$ in addition. On this basis, we conclude that the envelope solitons considered here lead to bound-state resonances and associated effects in a classical treatment. Moreover, their semiclassical quantization gives remarkably accurate energy levels.

Physical review. B, Condensed matter•Nov 1, 1981•62 citations
The easy-plane ferromagnetic chain in an applied magnetic field is known to map onto a sine-Gordon field theory, in the classical continuum limit. The sine-Gordon field theory supports breather excitations which, when quantised, are identifiable as a hierarchy of bound states of several small-amplitude excitations. For increasing values of the interaction parameter, the highest members of the hierarchy become unstable to the emission of a soliton/anti-soliton pair. For a critical value of the interaction parameter, even the ground state of the quantised sine-Gordon theory becomes unstable. For the quantum spin system, the excitations corresponding to the two lowest members of the hierarchy of breathers are the single-spin-wave and the two-spin-wave bound states. The authors examine the two-spin-wave bound-state excitations of the quantum spin system directly, using a large-S approach. Although the method does rely on large S, they recover previously derived results valid for arbitrary S, such as the exact two-spin-wave bound states for the isotropic Heisenberg ferromagnet and the expansion in the anisotropy about this result. In the limit of large anisotropy the two-spin-wave bound state becomes unstable, in a manner suggestive of the instability of the quantum-sine-Gordon field theory.
A variational estimate of the free energy of a quantum spin chain based on the stochastic behavior of the quantization axes is proposed. It allows a description of the nonlinear excitations and quantum spin waves within the same framework. For an easy-plane ferromagnet with an external field in the plane it reveals that the nonlinear excitations (solitons) are predominantly described by the classical degrees of freedom. Considerable effort has been devoted to the theoretical description1 and experimental verification of the existence of nonlinear excitations. In this field magnetic chain systems have played an important part.2 Chain systems with an easy-plane anisotropy, such as (C6 H11 NH3 )CuBr3 (CHAB) and CsNiF3 , which are subjected to a symmetry breaking field, were found to be soliton bearing.3 Theoretically the description of the nonlinear effect is based on the mapping of the spin dynamics on a sine-Gordon model. Further research4 indicated that the interpretation of the thermodynamical properties of real systems in terms of the sine-Gordon model required however quantum and anisotropy corrections. Since the dipolar interactions between the spins in the chain are considered to be negligible, the direction of the spin chain in configuration space with respect to any preferred direction in spin space does not enter explicitly in the Hamiltonian. This makes it interesting to define a site-dependent z direction in spin space which makes an angle (Θn ,φn ) with the z direction in configuration space. The model Hamiltonian can then be written in terms of the spin operators defined with respect to the site-dependent quantization axis. The representation with equal quantization axis for all the sites is related to the representation with different quantization axes at each site by a rotation in spin space. The unitary transformation that describes the rotation contains the angles (Θn ,φn ), which we have treated as random variables. Based on this observation we have proposed a new variational approach for the quantum spin chain. We will outline the main results here. A full account is published elsewhere.5 In our approach the free energy of the system can be separated in a quantum part and a classical part. The free energy F of the chain is then approximated by an upper bound F<Fcl +Fqm , where βFcl =−ln[E(exp−βVcl )] and βFqm =−ln Tr(exp−β〈Hqm 〉). The first part is a configurational classical free energy, depending on the choice of the probability distribution for the quantization axes via the expectation E. The second contribution is the free energy of a modified quantum chain, whose parameters are classical averages and can be obtained from the configurational classical free energy. For easy-plane ferromagnetic chains, the nonlinear excitations are comprised in the classical part, whereas the quantum part contains predominantly linear excitations. For those thermodynamical quantities that are insensitive to the contribution of the quantum excitations, the chain is thermodynamically equivalent with a classical sine-Gordon system. The out-of-plane corrections are incorporated in the quantum part of the free energy. This behavior is corroborated experimentally by measurements of the excess heat capacity of CHAB for different magnetic fields with the same in-plane component.6 The validity of our approach is, however, not restricted to easy-plane ferromagnetic chains. Also for more complex systems with, for example, an orthorhombic or helical symmetry, the introduction of stochastic quantization axes can shed new light on some of the thermodynamical properties of these quantum systems in terms of their classical equivalent.
Abstract A spin-1/2 Heisenberg antiferromagnetic chain is one of the most important paradigmatic models in quantum magnetism. Its ground state is a spin singlet, while the excitation spectrum is formed by gapless fractional excitations, spinons. The presence of alternating g-tensors and/or the staggered Dzyaloshinskii-Moriya interaction results in opening the energy gap Δ∝H2/3\varDelta \propto H^{2/3}Δ∝H2/3 Δ ∝ H 2 / 3 , once the magnetic field H is applied. A fairly good understanding of this phenomenon was achieved in the framework of the sine-Gordon quantum-field theory, taking into account the effective transverse staggered field induced by the applied uniform field. The theory predicts solitons and antisolitons as elementary excitations, as well as their bound states, breathers. Here, I review recent high-field electron spin resonance spectroscopy studies of such systems, focusing on peculiarities of their spin dynamics in the sine-Gordon regime and beyond.
We investigate the quantum corrections to the energy of the static soliton in the XY\mathrm{XY}XY-like ferromagnetic spin chain in a symmetry-breaking external field. We apply a semiclassical continuum approximation which allows us to treat the leading-order deviations from idealized planar behavior. The equivalent quantum-field theory, an extension of the quantum sine-Gordon theory, is renormalized and its soliton energy is shown to depend strongly on this deviation. The quantum-field-theoretical renormalization, however, does not affect the energy of the magnetic soliton, which is calculated to leading order in 1S\frac{1}{S}S1​, where SSS is the spin length. The sine-Gordon limit is found applicable only for very large anisotropies; for realistic anisotropies, the reduction of the soliton energy owing to quantum effects is enhanced and depends only on SSS. A quantitative calculation for the S=1S=1S=1 system CsNiF3{\mathrm{F}}_{3}F3​ is likely to require higher-order terms in 1S\frac{1}{S}S1​.
Spinon, soliton and breather in the spin-1/2 antiferromagnetic chain KCuGaF6_66​
Elementary excitations of the S=1/2 one-dimensional antiferromagnet KCuGaF_6 were investigated by inelastic neutron scattering in zero and finite magnetic fields perpendicular to the (1, 1, 0) plane combined with specific heat measurements. KCuGaF6_66​ exhibits no long-range magnetic ordering down to 50 mK despite the large exchange interaction J/k_B=103 K. At zero magnetic field, well-defined spinon excitations were observed. The energy of the des Cloizeaux and Pearson mode of the spinon excitations is somewhat larger than that calculated with the above exchange constant. This discrepancy is mostly ascribed to the effective XY anisotropy arising from the large Dzyaloshinsky-Moriya interaction with an alternating D vector. KCuGaF_6 in a magnetic field is represented by the quantum sine-Gordon model, for which low-energy elementary excitations are composed of solitons, antisolitons and their bound states called breathers. Unlike the theoretical prediction, it was found that the energy of a soliton is smaller than that of the first breather, although the energy of the first breather coincides with that observed in a previous ESR measurement.
We investigate the relation between magnon, magnon bound states, and the classical soliton solutions in the isotropic, the anisotropic-exchange, and the easy-axis ferromagnetic Heisenberg chains. The Dyson-Maleev boson representation is used to study the interaction between magnons, and bound states are investigated in terms of two-spin Green's functions. For easy-axis and anisotropic-exchange magnets, a mapping to the Bose gas with attractive $\ensuremath{\delta}$-function interactions is established, and it yields the eigenvalues of the magnon and the mmmth bound state in the weak-coupling and continuum limits. The classical limit of the two-spin Green's function reveals that, to leading order in temperature, the bound-state resonance and the associated effects on the two-magnon continuum survives. An important result is a T2{T}^{2}T2 dependence of the "binding" energy of the bound-state resonance. As a consequence, in a classical description the bound states enter only in order T2{T}^{2}T2. Finally, we quantize the soliton solutions according to the Bohr-Sommerfeld and de Broglie rules. This approach is found to be exact for the s=12s=\frac{1}{2}s=21​ isotropic Heisenberg chain and for sufficiently small wave numbers for all sss values. In the anisotropic-exchange and easy-axis models, it agrees with the results obtained from the mapping on the Bose gas, for large quantum numbers and in the easy-axis case for $s\ensuremath{\gg}\frac{1}{2}$ in addition. On this basis, we conclude that the envelope solitons considered here lead to bound-state resonances and associated effects in a classical treatment. Moreover, their semiclassical quantization gives remarkably accurate energy levels.