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arXiv:cond-mat/9812296 · The European Physical Journal B

Incommensurability in the magnetic excitations of the bilinear-biquadratic spin-1 chain

O. Golinelli, Th. Jolicœur, Erik Schwartz Sørensen

Published Jan 1, 1999• 22 citations
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Abstract

We study the magnetic excitation spectrum of the spin-1 chain with Hamiltonian H = Σ i cos θ S i · S i+1 + sin θ(S i · S i+1)2. We focus on the range 0 ≤ θ ≤ + π/4 where the spin chain is in the gapped Haldane phase. The excitation spectrum and static structure factor is studied using direct Lanczos diagonalization of small systems and density-matrix renormalization group techniques combined with the single-mode approximation. The magnon dispersion has a minimum at k = π until a critical value θ c = 0.38 is reached at which the curvature (velocity) vanishes. Beyond this point, which is distinct from the VBS point and the Lifshitz point, the minimum lies at an incommensurate value that goes smoothly to k = 2π/3 when θ approaches π/4, the Lai-Sutherland point. The mode remains isolated from the other states: there is no evidence of spinon deconfinement before the point θ = +π/4. These findings explain recent observation of the behavior of the magnetization curve M ≈ (H − H c)1/4 for θ = θ c.

DOI
10.1007/bf03219165

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Critical behavior of the S=1 antiferromagnetic Heisenberg chain (abstract)

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The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.

Journal of Applied Physics•Apr 15, 1991•1 citation
We study the magnetic excitation spectrum of the spin-1 chain with Hamiltonian H = Σ i cos θ S i · S i+1 + sin θ(S i · S i+1)2. We focus on the range 0 ≤ θ ≤ + π/4 where the spin chain is in the gapped Haldane phase. The excitation spectrum and static structure factor is studied using direct Lanczos diagonalization of small systems and density-matrix renormalization group techniques combined with the single-mode approximation. The magnon dispersion has a minimum at k = π until a critical value θ c = 0.38 is reached at which the curvature (velocity) vanishes. Beyond this point, which is distinct from the VBS point and the Lifshitz point, the minimum lies at an incommensurate value that goes smoothly to k = 2π/3 when θ approaches π/4, the Lai-Sutherland point. The mode remains isolated from the other states: there is no evidence of spinon deconfinement before the point θ = +π/4. These findings explain recent observation of the behavior of the magnetization curve M ≈ (H − H c)1/4 for θ = θ c.
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 $(J\mathrm{cos}\ensuremath{\theta})$ and biquadratic $(J\mathrm{sin}\ensuremath{\theta})$ exchange are investigated via the recursion method for systems with up to N=18N=18N=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=0T=0T=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 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 $q=\ensuremath{\pi}$ are found to consist of single modes with NNN-independent excitation energies ${\ensuremath{\omega}}_{\ensuremath{\lambda}}^{D}/|{e}_{0}|=5$ and ${\ensuremath{\omega}}_{\ensuremath{\lambda}}^{T}/|{e}_{0}|=6,$ 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 $(\ensuremath{\theta}=\ensuremath{-}\ensuremath{\pi}/4),$ the two-magnon excitations turn into two-spinon excitations.
We investigate spin chains with bilinear-biquadratic (BLBQ) spin interactions as a function of an applied magnetic field hhh. At the Uimin-Lai-Sutherland (ULS) critical point we find a gapless to gapless transition revealed by the dynamical structure factor $S(q,\ensuremath{\omega})$ as a function of hhh. At h=0h=0h=0, the envelope of the lowest-energy excitations goes soft at two points, ${q}_{1}=2\ensuremath{\pi}/3$ and ${q}_{2}=4\ensuremath{\pi}/3$, dubbed the phase A. With increasing field, the spectral peaks at each of the gapless points bifurcate, making in total four soft modes, and combine to form a new set of excitations that soften at a single point $q=\ensuremath{\pi}$ at ${h}_{c1}\ensuremath{\approx}0.94$. Beyond hc1{h}_{c1}hc1​ the system enters another gapless B phase until the transition at hc2=4{h}_{c2}=4hc2​=4 to the fully polarized phase. We compare the ULS model results with those for the Affleck-Kennedy-Lieb-Tasaki model as a representative of the gapped Haldane phase. We explain the mechanism of the gapless to gapless transition in the ULS model using its conserved charges and a spinon band picture. We also discuss the universality of central charges of the BLBQ family of models subjected to a magnetic field.
We investigate spin chains with bilinear-biquadratic spin interactions as a function of an applied magnetic field hhh. At the Uimin-Lai-Sutherland (ULS) critical point we find a remarkable hierarchy of fractionalized excitations revealed by the dynamical structure factor S(q,ω)S(q,\omega)S(q,ω) as a function of magnetic field yielding a transition from a gapless phase to another gapless phase before reaching the fully polarized state. At h=0h=0h=0, the envelope of the lowest energy excitations goes soft at two points q1=2π/3q_1=2\pi/3q1​=2π/3 and q2=4π/3q_2=4\pi/3q2​=4π/3, dubbed the A-phase. With increasing field, the spectral peaks at each of the gapless points bifurcate and combine to form a new set of fractionalized excitations that soften at a single point q=πq=\piq=π at hc1≈0.94h_{c1}\approx 0.94hc1​≈0.94. Beyond hc1h_{c1}hc1​ the system remains in this phase dubbed the B-phase until the transition at hc2=4h_{c2}=4hc2​=4 to the fully polarized phase. We discuss the central charge of these two gapless phases and contrast the behavior with that of the gapped Haldane phase in a field.
Using a time-dependent density matrix renormalization group approach, we study the dynamical structure factor of the ${J}_{1}\ensuremath{-}{J}_{2}$ spin-1 chain. As J2{J}_{2}J2​ increases, the magnon mode develops incommensurability. The system undergoes a first-order transition at J2=0.76J1{J}_{2}=0.76\phantom{\rule{0.16em}{0ex}}{J}_{1}J2​=0.76J1​, and at that point, domain walls lead to a continuum of fractional quasiparticles or spinons. By studying small variations in J2{J}_{2}J2​ around the transition point, we observe the confinement of spinons into bound states in the spectral function and find a smooth evolution of the spectrum into magnon modes away from the phase transition. We employ the single-mode approximation to accurately account for the dispersion of the magnon mode away from the phase transition and describe the associated continua and bound states. We extend the single-mode approximation to describe the dispersion of a spinon at the phase transition point and obtain its dispersion throughout the Brillouin zone. This allows us to relate the incommensurability at and around the transition point to the competition between a negative nearest-neighbor hopping amplitude and a positive next-nearest-neighbor one for the domain wall.
The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.