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Journal of Physics A Mathematical and General

Gapless excitations in the spin-1 bilinear-biquadratic chain

P Reed

Published Feb 7, 1994• 8 citations
Publisher

Abstract

The susceptibility and real part of the structure factor have been calculated for the spin-1 bilinear-biquadratic chain using Trotter-Suzuki decomposition and transfer matrix methods. The results indicate an extended region around the Lai-Sutherland point where the excitations are gapless. This result is in agreement with direct diagonalization for small clusters but is in conflict with a recent spin-wave analysis.

DOI
10.1088/0305-4470/27/3/002

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The susceptibility and real part of the structure factor have been calculated for the spin-1 bilinear-biquadratic chain using Trotter-Suzuki decomposition and transfer matrix methods. The results indicate an extended region around the Lai-Sutherland point where the excitations are gapless. This result is in agreement with direct diagonalization for small clusters but is in conflict with a recent spin-wave analysis.
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 study numerically the elementary excitation spectrum of the most general isotropic spin-1 chain with bilinear-biquadratic nearest-neighbor coupling. Using finite-size scaling, a massless phase with a period tripling in the ground state is found to exist in an extended region around the Lai-Sutherland point. The location of the transition from the known valence-bond-like phase to the trimerized phase cannot, however, be given precisely. We also present results on the ground-state two-point correlation function.
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.
Measurements done on LiVGe2O6 by Millet et al. [Phys. Rev. Lett. 83, 4176 (1999)] on phases of the Singlet ground state of the Spin-1 Chain can be described by the Bilinear-biquadratic spin-1 chain. I will discuss these results, among the results of others and talk about theoretical calculations done by A. Lauchli [Phys. Rev. B 74, 144426 (2006)] that describe the phase diagram of the bilinear-biquadratic model. At a particular point in the phase diagram the model becomes SU(3) globally symmetric and is known as the Lai-Sutherland model, it is Bethe ansatz solvable as shown in [A Schmitt et al., J. Phys. A: Math. Gen. 29 (1996) 3951-3962]. Finally I will describe the thermodynamics of the model as where calculated by Lou et al. [Phys. Rev. Lett. 85, 11 (2000)].
We have studied the Spin 1 antiferromagnetic chain with both bilinear and biquadratic exchange interactions. The spin correlations have been calculated by means of the transfer matrix method based on the Trotter formula. The mass gap ( m (0)) at the ground state has been investigated as a function of the ratio λ between bilinear and biquadratic interactions. The result suggests that the gapless ground state persists in a finite range of λ, i.e., λ c < λ< λ 1 , with λ c = 0.5 and λ 1 > 1 which is consistent with the exact solution at the completely integrable point (λ= 1). The calculated mass gap ( m (0)) has been fitted to a Kosterlitz-Thouless (KT)-type for λ< λ c ( m (0)= A exp [- B /(λ c - λ) σ ]) producing the results σ= 0.49±0.04 and λ c =0.49±0.03.
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.