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Phases of the Bilinear-biquadratic Spin-1 Chains

Juan Alberto Garcia

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

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)].

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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)].
The magnetic susceptibility and specific heat of the one-dimensional S = 1 bilinear-biquadratic Heisenberg model are calculated using the transfer matrix renormalization group. By comparing the results with the experimental data of LiVGe2O6 measured by Millet et al. [Phys. Rev. Lett. 83, 4176 (1999)], we find that the susceptibility data of this material, after subtracting the impurity contribution, can be quantitatively explained with this model. The biquadratic exchange interaction in this material is found to be ferromagnetic, i.e., with a positive coupling constant.
The one-dimensional S = 1 quantum spin chain with a linear and bilinear nearest-neighbour interaction of equal strength is integrable by the nested Bethe ansatz. In this paper the model is studied in the presence of an external magnetic field and an internal crystal field. By solving the Bethe ansatz equations for chains up to length N = 360 we construct the complete phase diagram of the system. We discuss how the magnetization curves depend on the internal field. In the SU(3) phase, where all three densities of atoms with -component -1, 0, +1 are non-vanishing, lines of constant density can be approximately parametrized by modified hypocycloides.
The possible ground states and the phase transitions of a spin-1 chain with bilinear-biquadratic exchange are studied numerically. In order to see better the real asymptotic behavior of the infinite system and to determine the necessary degeneracies, a twisted boundary condition is used that reduces the finite-size effects. This calculation confirms that a second-order Ising-type phase transition occurs at the Takhtajan-Babujian point from the dimerized phase to the singlet Haldane phase. The transition at the Lai-Sutherland point, to a massless phase with three soft modes, is found to be of Kosterlitz-Thouless type.
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.