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

Static and dynamic structure factors in the Haldane phase of the bilinear-biquadratic spin-1 chain

Andreas Schmitt, K.-H. Mütter, Michael Karbach, Yongmin Yu, Gerhard H. Muller

Published Sep 1, 1998• 28 citations
Publisher

Abstract

The excitation spectra of the $T=0$ dynamic structure factors for the spin, dimer, and trimer fluctuation operators as well as for the center fluctuation operator in the one-dimensional $S=1$ Heisenberg model with isotropic bilinear $(J\mathrm{cos}\ensuremath{\theta})$ and biquadratic $(J\mathrm{sin}\ensuremath{\theta})$ exchange are investigated via the recursion method for systems with up to $N=18$ sites over the predicted range, $\ensuremath{-}\ensuremath{\pi}/4<\ensuremath{\theta}\ensuremath{\lesssim}\ensuremath{\pi}/4,$ of the topologically ordered Haldane phase. The four static and dynamic structure factors probe the ordering tendencies in the various coupling regimes and the elementary and composite excitations which dominate the $T=0$ dynamics. At $\ensuremath{\theta}=\mathrm{arctan}\frac{1}{3}$ (valence-bond solid point), the dynamically relevant spectra in the invariant subspaces with total spin ${S}_{T}=0,1,2$ are dominated by a branch of magnon states ${(S}_{T}=1),$ by continua of two-magnon scattering states ${(S}_{T}=0,1,2),$ and by discrete branches of two-magnon bound states with positive interaction energy ${(S}_{T}=0,2).$ The dimer and trimer spectra at $q=\ensuremath{\pi}$ are found to consist of single modes with $N$-independent excitation energies ${\ensuremath{\omega}}_{\ensuremath{\lambda}}^{D}/|{e}_{0}|=5$ and ${\ensuremath{\omega}}_{\ensuremath{\lambda}}^{T}/|{e}_{0}|=6,$ where ${e}_{0}{=E}_{0}/N$ is the ground-state energy per site. The basic structure of the dynamically relevant excitation spectrum remains the same over a substantial parameter range within the Haldane phase. At the transition to the dimerized phase $(\ensuremath{\theta}=\ensuremath{-}\ensuremath{\pi}/4),$ the two-magnon excitations turn into two-spinon excitations.

DOI
10.1103/physrevb.58.5498

Related papers

Static and dynamic structure factors in the Haldane phase of the bilinear-biquadratic spin-1

K.-H. Mütter, Schmitt, A, Michael Karbach +2 more

The excitation spectra of the T=0 dynamic structure factors for the spin, dimer, and trimer fluctuation operators as well as for the newly defined center fluctuation operator in the one-dimensional S=1 Heisenberg model wi th isotropic bilinear $(J\cosθ)$ and biquadratic $(J\sinθ)$ exchange are investigated via the recursion method for systems with up to N=18 site s over the predicted range, $-π/4

arXiv:cond-mat/9803147•Mar 12, 1998•0 citations

Low-energy physics of isotropic spin-1 chains in the critical and Haldane phases

Moritz Binder, Thomas Barthel

Using a matrix product state algorithm with infinite boundary conditions, we compute high-resolution dynamic spin and quadrupolar structure factors in the thermodynamic limit to explore the low-energy excitations of isotropic bilinear-biquadratic spin-1 chains. Haldane mapped the spin-1 Heisenberg antiferromagnet to a continuum field theory, the nonlinear sigma model ($\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$). We find that the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ fails to capture the influence of the biquadratic term and provides only an unsatisfactory description of the Haldane phase physics. But several features in the Haldane phase can be explained by noninteracting multimagnon states. The physics at the Uimin-Lai-Sutherland point is characterized by multisoliton continua. Moving into the extended critical phase, we find that these excitation continua contract, which we explain using a field-theoretic description. New excitations emerge at higher energies and, in the vicinity of the purely biquadratic point, they show simple cosine dispersions. Using block fidelities, we identify them as elementary one-particle excitations and relate them to the integrable Temperley-Lieb chain.

arXiv:2005.03643•Physical review. B./Physical review. B•Jul 27, 2020•11 citations

Dynamical structure factors of S=1 bond-alternating Heisenberg chains

Takahumi Suzuki, Sei-ichiro Suga

We calculate the dynamical structural factor of the $S=1$ bond-alternating Heisenberg chain. In the Haldane phase, the lowest excited states form the lower edge of the multimagnon continuum in $0<~q<~{q}_{c}$ and the one-magnon mode in ${q}_{c}<~q<~\ensuremath{\pi}.$ As the system approaches the gapless point, ${q}_{c}$ shifts towards $q=\ensuremath{\pi}$ and the largest integrated intensity of the one-magnon mode is decreased. In the singlet-dimer phase, the one-magnon mode appears in $0<~q<~{q}_{c}.$ As the bond-alternation becomes strong, ${q}_{c}$ shifts towards $q=\ensuremath{\pi}.$ In the antiferromagnetic-ferromagnetic bond-alternation region with a strong ferromagnetic coupling, the lowest excited states form the lower edge of the multimagnon continuum in $0<~q<~0.2\ensuremath{\pi}$ and $0.8\ensuremath{\pi}<~q<~\ensuremath{\pi},$ and the one-magnon mode appears in $0.2\ensuremath{\pi}<q<0.8\ensuremath{\pi}.$ The largest integrated intensity of the one-magnon mode is $93%,$ which is slightly smaller than that in the $S=1$ Haldane-gap system. We further discuss the dynamical structural factor in connection with the inelastic neutron-scattering experiments.

arXiv:cond-mat/0309178•Physical review. B, Condensed matter•Oct 17, 2003•5 citations

Confinement, deconfinement, and bound states in the spin-1 and spin- 3 2 generalizations of the Majumdar-Ghosh chain

Aman Sharma, Mithilesh Nayak, Natalia Chepiga +2 more

We investigate the nature of low-energy excitations in a spin chain with antiferrmomagnetic nearest-neighbor $J_1$, next-nearest-neighbor $J_2$, and three-site $J_3$ interactions using the time-dependent density matrix renormalization group and the single mode approximation techniques. In the absence of the $J_2$ interaction, we identify clear distinctions in the spectral functions in the fully dimerized phase across the exactly dimerized line for different magnitudes of the spins. In contrast to the spin-$1/2$ chain, where the spinon continuum dominates the spectral functions, the magnon modes are prominent in the spectral functions of the spin-$1$ and spin-$3/2$ chains. Through single mode approximation and valence bond solid approaches, we disentangle magnon and spinon contributions to the spectral functions. After including the $J_2$ interactions, for the spin-$1$ chain we trace the evolution of the dynamical structure factor along the phase transition line between the Haldane phase and the fully dimerized phase. We find that the excitation spectrum is a continuum along this line and the spectral gap closes as the order of the transition changes from first order to second order. Along the line of first-order transitions, the spinon-like domain walls are deconfined, and the model exhibits their confinement into discrete bound states away from the transition line. A similar phenomenon occurs in the spin-$3/2$ chain across the phase transition between partially dimerized to fully dimerized phases, revealing a universal spinon confinement phenomenon across first-order phase transitions. This study presents the dynamical structure factor corresponding to the ground state phase diagram and establishes a unified quasiparticle framework for understanding the fundamental nature of excitations across distinct quantum phases in frustrated $J_1$-$J_2$-$J_3$ Heisenberg spin chains.

arXiv:2509.06720•Physical review. B./Physical review. B•Feb 2, 2026•0 citations

Dynamics of quantum spin systems in dimer and valence-bond-solid ground states stabilized by competing interactions

Yongmin Yu, Gerhard H. Muller

For special coupling ratios, the one-dimensional (1D) s=1/2 Heisenberg model with antiferromagnetic nearest and next-nearest neighbor interactions has a pure dimer ground state, and the 1D s=1 Heisenberg model with antiferromagnetic bilinear and biquadratic interactions has an exact valence-bond-solid ground state. The recursion method is used to calculate the T=0 spin dynamic structure factor for both models and, for the s=1/2 model, also the dimer dynamic structure factor. New results for line shapes and dynamically relevant dispersions are obtained.

arXiv:cond-mat/9508124•Journal of Applied Physics•Apr 15, 1996•2 citations

Excitations and dynamical structure factor of J 1 − J 2 spin- 3 2 and spin- 5 2 Heisenberg spin chains

Aman Sharma, Mithilesh Nayak, Natalia Chepiga +1 more

We study the dynamical structure factor of the frustrated spin-$3/2\phantom{\rule{4pt}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg chains, with particular focus on the partially dimerized phase that emerges between two Kosterlitz-Thouless transitions. Using a valence bond solid Ansatz corroborated by density-matrix renormalization-group simulations, we investigate the nature of magnon and spinon excitations through the single-mode approximation. We show that the magnon develops an incommensurate dispersion at ${J}_{2}\ensuremath{\approx}0.32{J}_{1}$, while the spinons, viewed as domain walls between degenerate valence bond solid states, become incommensurate at ${J}_{2}\ensuremath{\approx}0.4{J}_{1}$ beyond the Lifshitz point $({J}_{2}\ensuremath{\approx}0.388{J}_{1})$. The dynamical structure factor exhibits rich spectral features shaped by the interplay between these excitations, with magnons appearing as resonances embedded in the spinon continuum. The spinon gap shows a nonmonotonic behavior, reaching a peak near the center of the partially dimerized phase and closing at the boundaries, suggesting the appearance of a floating phase as a result of the condensation of incommensurate spinons. Comparative analysis with the spin-$5/2$ case confirms the universality of these phenomena across half-integer higher-spin systems. Our results provide detailed insight into how fractionalization and incommensurate condensation govern the spectral properties of frustrated spin chains, offering a unified picture across different spin magnitudes.

arXiv:2506.13431•Physical review. B./Physical review. B•Aug 15, 2025•1 citation
The excitation spectra of the T=0T=0T=0 dynamic structure factors for the spin, dimer, and trimer fluctuation operators as well as for the center fluctuation operator in the one-dimensional S=1S=1S=1 Heisenberg model with isotropic bilinear (Jcos\ensuremathθ)(J\mathrm{cos}\ensuremath{\theta})(Jcos\ensuremathθ) and biquadratic (Jsin\ensuremathθ)(J\mathrm{sin}\ensuremath{\theta})(Jsin\ensuremathθ) exchange are investigated via the recursion method for systems with up to N=18N=18N=18 sites over the predicted range, \ensuremath−\ensuremathπ/4<\ensuremathθ\ensuremath≲\ensuremathπ/4,\ensuremath{-}\ensuremath{\pi}/4<\ensuremath{\theta}\ensuremath{\lesssim}\ensuremath{\pi}/4,\ensuremath−\ensuremathπ/4<\ensuremathθ\ensuremath≲\ensuremathπ/4, of the topologically ordered Haldane phase. The four static and dynamic structure factors probe the ordering tendencies in the various coupling regimes and the elementary and composite excitations which dominate the T=0T=0T=0 dynamics. At \ensuremathθ=arctan13\ensuremath{\theta}=\mathrm{arctan}\frac{1}{3}\ensuremathθ=arctan31​ (valence-bond solid point), the dynamically relevant spectra in the invariant subspaces with total spin ST=0,1,2{S}_{T}=0,1,2ST​=0,1,2 are dominated by a branch of magnon states (ST=1),{(S}_{T}=1),(ST​=1), by continua of two-magnon scattering states (ST=0,1,2),{(S}_{T}=0,1,2),(ST​=0,1,2), and by discrete branches of two-magnon bound states with positive interaction energy (ST=0,2).{(S}_{T}=0,2).(ST​=0,2). The dimer and trimer spectra at q=\ensuremathπq=\ensuremath{\pi}q=\ensuremathπ are found to consist of single modes with NNN-independent excitation energies \ensuremathω\ensuremathλD/∣e0∣=5{\ensuremath{\omega}}_{\ensuremath{\lambda}}^{D}/|{e}_{0}|=5\ensuremathω\ensuremathλD​/∣e0​∣=5 and \ensuremathω\ensuremathλT/∣e0∣=6,{\ensuremath{\omega}}_{\ensuremath{\lambda}}^{T}/|{e}_{0}|=6,\ensuremathω\ensuremathλT​/∣e0​∣=6, where e0=E0/N{e}_{0}{=E}_{0}/Ne0​=E0​/N is the ground-state energy per site. The basic structure of the dynamically relevant excitation spectrum remains the same over a substantial parameter range within the Haldane phase. At the transition to the dimerized phase (\ensuremathθ=\ensuremath−\ensuremathπ/4),(\ensuremath{\theta}=\ensuremath{-}\ensuremath{\pi}/4),(\ensuremathθ=\ensuremath−\ensuremathπ/4), the two-magnon excitations turn into two-spinon excitations.
The excitation spectra of the T=0 dynamic structure factors for the spin, dimer, and trimer fluctuation operators as well as for the newly defined center fluctuation operator in the one-dimensional S=1 Heisenberg model wi th isotropic bilinear (Jcos⁡θ)(J\cosθ)(Jcosθ) and biquadratic (Jsin⁡θ)(J\sinθ)(Jsinθ) exchange are investigated via the recursion method for systems with up to N=18 site s over the predicted range, $-π/4
Using a matrix product state algorithm with infinite boundary conditions, we compute high-resolution dynamic spin and quadrupolar structure factors in the thermodynamic limit to explore the low-energy excitations of isotropic bilinear-biquadratic spin-1 chains. Haldane mapped the spin-1 Heisenberg antiferromagnet to a continuum field theory, the nonlinear sigma model ($\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$). We find that the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ fails to capture the influence of the biquadratic term and provides only an unsatisfactory description of the Haldane phase physics. But several features in the Haldane phase can be explained by noninteracting multimagnon states. The physics at the Uimin-Lai-Sutherland point is characterized by multisoliton continua. Moving into the extended critical phase, we find that these excitation continua contract, which we explain using a field-theoretic description. New excitations emerge at higher energies and, in the vicinity of the purely biquadratic point, they show simple cosine dispersions. Using block fidelities, we identify them as elementary one-particle excitations and relate them to the integrable Temperley-Lieb chain.
We calculate the dynamical structural factor of the S=1S=1S=1 bond-alternating Heisenberg chain. In the Haldane phase, the lowest excited states form the lower edge of the multimagnon continuum in 0< q< qc0<~q<~{q}_{c}0< q< qc​ and the one-magnon mode in ${q}_{c}<~q<~\ensuremath{\pi}.$ As the system approaches the gapless point, qc{q}_{c}qc​ shifts towards $q=\ensuremath{\pi}$ and the largest integrated intensity of the one-magnon mode is decreased. In the singlet-dimer phase, the one-magnon mode appears in 0< q< qc.0<~q<~{q}_{c}.0< q< qc​. As the bond-alternation becomes strong, qc{q}_{c}qc​ shifts towards $q=\ensuremath{\pi}.$ In the antiferromagnetic-ferromagnetic bond-alternation region with a strong ferromagnetic coupling, the lowest excited states form the lower edge of the multimagnon continuum in $0<~q<~0.2\ensuremath{\pi}$ and $0.8\ensuremath{\pi}<~q<~\ensuremath{\pi},$ and the one-magnon mode appears in $0.2\ensuremath{\pi}<q<0.8\ensuremath{\pi}.$ The largest integrated intensity of the one-magnon mode is 9393%,93 which is slightly smaller than that in the S=1S=1S=1 Haldane-gap system. We further discuss the dynamical structural factor in connection with the inelastic neutron-scattering experiments.
We investigate the nature of low-energy excitations in a spin chain with antiferrmomagnetic nearest-neighbor J1J_1J1​, next-nearest-neighbor J2J_2J2​, and three-site J3J_3J3​ interactions using the time-dependent density matrix renormalization group and the single mode approximation techniques. In the absence of the J2J_2J2​ interaction, we identify clear distinctions in the spectral functions in the fully dimerized phase across the exactly dimerized line for different magnitudes of the spins. In contrast to the spin-1/21/21/2 chain, where the spinon continuum dominates the spectral functions, the magnon modes are prominent in the spectral functions of the spin-111 and spin-3/23/23/2 chains. Through single mode approximation and valence bond solid approaches, we disentangle magnon and spinon contributions to the spectral functions. After including the J2J_2J2​ interactions, for the spin-111 chain we trace the evolution of the dynamical structure factor along the phase transition line between the Haldane phase and the fully dimerized phase. We find that the excitation spectrum is a continuum along this line and the spectral gap closes as the order of the transition changes from first order to second order. Along the line of first-order transitions, the spinon-like domain walls are deconfined, and the model exhibits their confinement into discrete bound states away from the transition line. A similar phenomenon occurs in the spin-3/23/23/2 chain across the phase transition between partially dimerized to fully dimerized phases, revealing a universal spinon confinement phenomenon across first-order phase transitions. This study presents the dynamical structure factor corresponding to the ground state phase diagram and establishes a unified quasiparticle framework for understanding the fundamental nature of excitations across distinct quantum phases in frustrated J1J_1J1​-J2J_2J2​-J3J_3J3​ Heisenberg spin chains.
For special coupling ratios, the one-dimensional (1D) s=1/2 Heisenberg model with antiferromagnetic nearest and next-nearest neighbor interactions has a pure dimer ground state, and the 1D s=1 Heisenberg model with antiferromagnetic bilinear and biquadratic interactions has an exact valence-bond-solid ground state. The recursion method is used to calculate the T=0 spin dynamic structure factor for both models and, for the s=1/2 model, also the dimer dynamic structure factor. New results for line shapes and dynamically relevant dispersions are obtained.
We study the dynamical structure factor of the frustrated spin-$3/2\phantom{\rule{4pt}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg chains, with particular focus on the partially dimerized phase that emerges between two Kosterlitz-Thouless transitions. Using a valence bond solid Ansatz corroborated by density-matrix renormalization-group simulations, we investigate the nature of magnon and spinon excitations through the single-mode approximation. We show that the magnon develops an incommensurate dispersion at ${J}_{2}\ensuremath{\approx}0.32{J}_{1}$, while the spinons, viewed as domain walls between degenerate valence bond solid states, become incommensurate at ${J}_{2}\ensuremath{\approx}0.4{J}_{1}$ beyond the Lifshitz point $({J}_{2}\ensuremath{\approx}0.388{J}_{1})$. The dynamical structure factor exhibits rich spectral features shaped by the interplay between these excitations, with magnons appearing as resonances embedded in the spinon continuum. The spinon gap shows a nonmonotonic behavior, reaching a peak near the center of the partially dimerized phase and closing at the boundaries, suggesting the appearance of a floating phase as a result of the condensation of incommensurate spinons. Comparative analysis with the spin-5/25/25/2 case confirms the universality of these phenomena across half-integer higher-spin systems. Our results provide detailed insight into how fractionalization and incommensurate condensation govern the spectral properties of frustrated spin chains, offering a unified picture across different spin magnitudes.