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arXiv:cond-mat/0003102 · Physical Review Letters

Thermodynamics of the Bilinear-Biquadratic Spin-One Heisenberg Chain

Jizhong Lou, Tao Xiang, Zhao-Bin Su

Published Sep 11, 2000• 43 citations
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Abstract

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.

DOI
10.1103/physrevlett.85.2380

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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.
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)].
Thermodynamic properties of the quantum Heisenberg spin chains with S=1/2,S=1/2,S=1/2, 1, and 3/23/23/2 are investigated using the transfer-matrix renormalization-group method. The temperature dependence of the magnetization, susceptibility, specific heat, spin-spin correlation length, and several other physical quantities in a zero or finite applied field are calculated and compared. Our data agree well with the Bethe ansatz, exact diagonalization, and quantum Monte Carlo results and provide further insight into the quantum effects in the antiferromagnetic Heisenberg spin chains.
The authors present calculations of the specific heat of 1D quantum spin chains for a number of integrable and non-integrable Hamiltonians, in the presence of a magnetic field. For S=1 the Hamiltonian has both bilinear and biquadratic isotropic exchange and includes ferromagnetic and antiferromagnetic versions of the Heisenberg, Takhtajan-Babujian (TB), Lai-Sutherland (LS) (permutation) and biquadratic models. The two methods used are (i) direct numerical diagonalisation for chains of up to nine atoms and (ii) numerical solution of the Bethe ansatz equations for the integrable models (TB and LS). Generalisations of the latter method to S>1 are also given. For the integrable models the two methods give results in good agreement. Various types of behaviour, particularly at low T, are found and these are related to known phase changes at T=0. The results of the first method for the non-integrable Hamiltonians are discussed with reference to the existence of the Haldane gap in the S=1 Heisenberg model and a possible gap in the biquadratic model.
The density matrix renormalization group (DMRG) method is applied to the anisotropic Heisenberg chain at finite temperatures. The free energy of the system is obtained using the quantum transfer matrix which is iteratively enlarged in the imaginary time direction. The magnetic susceptibility and the specific heat are calculated down to T=0.01J and compared with the Bethe ansatz results. The agreement including the logarithmic correction in the magnetic susceptibility at the isotropic point is fairly good.
Thermodynamic properties of the one-dimensional Kondo lattice model at half-filling are studied by the density matrix renormalization group method applied to the quantum transfer matrix. Spin susceptibility, charge susceptibility, and specific heat are calculated down to T=0.1t for various exchange constants. The obtained results clearly show crossover behavior from the high temperature regime of nearly independent localized spins and conduction electrons to the low temperature regime where the two degrees of freedom couple strongly. The low temperature energy scales of the charge and spin susceptibilities are determined and shown to be equal to the quasiparticle gap and the spin gap, respectively, for weak exchange couplings.
The magnetic susceptibility of a new one-dimensional S=1S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1S=1 system, the vanadium oxide LiVGe2O6{\mathrm{LiVGe}}_{2}{\mathrm{O}}_{6}LiVGe2​O6​, has been measured. Contrary to previous S=1S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1S=1 chains, it exhibits an abrupt drop at 22 K typical of a spin-Peierls transition, and it is consistent with a gapless spectrum above this temperature. We propose that this behavior is due to the presence of a significant biquadratic exchange interaction, a suggestion supported by quantum chemistry calculations that take into account the quasidegeneracy of the t2g{t}_{2g}t2g​ levels.