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Journal of Physics A Mathematical and General

Conformal anomaly and critical exponents of the spin-1 Takhtajan-Babujian model

Francisco Castilho Alcaraz, Marcio Jose Martins

Published Apr 7, 1988• 34 citations
Publisher

Abstract

The authors consider the critical behaviour of the spin-1 Hamiltonian of the Takhtajan-Babujian model. The Bethe ansatz equations for this antiferromagnet in a finite chain are investigated analytically and numerically. The conformal anomaly and critical exponents are obtained by exploiting their relations with the eigenspectrum of the finite system. The results strongly support the conjecture that the Wess-Zumino-Witten non-linear sigma model with topological charge k=2 is the underlying field theory for this statistical mechanics model.

DOI
10.1088/0305-4470/21/7/004

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The authors consider the critical behaviour of the spin-1 Hamiltonian of the Takhtajan-Babujian model. The Bethe ansatz equations for this antiferromagnet in a finite chain are investigated analytically and numerically. The conformal anomaly and critical exponents are obtained by exploiting their relations with the eigenspectrum of the finite system. The results strongly support the conjecture that the Wess-Zumino-Witten non-linear sigma model with topological charge k=2 is the underlying field theory for this statistical mechanics model.
The authors are concerned with the critical properties of antiferromagnetic Takhtajan-Babujian models with spin S=1, 3/2 and 2. The leading eigenenergies of this Hamiltonian, in a finite chain, are calculated by investigating numerically and analytically the Bethe ansatz equations for the finite system. The critical exponents and the conformal anomaly are obtained from their relations with the eigenspectrum of the finite Hamiltonian. The appearance of logarithmic corrections produces poor estimates. However, a combination of analytical and numerical methods produces very good estimates. Their results strongly support the conjecture that the Wess-Zumino-Witten-Novikov non-linear sigma models with topological charge k=2S are the underlying field theories for these spin-S statistical mechanics models.
The eigenenergies for the spin SSS generalization of the critical XXZ\mathrm{XXZ}XXZ model are calculated by solving numerically, for finite chains, their associated Bethe-Ansatz equations. The conformal anomaly and scaling dimensions are obtained from the conformal invariance of the infinite chain. Our results reveal that these models are governed by a conformal field theory with central charge c=3S(S+1)c=\frac{3S}{(S+1)}c=(S+1)3S​ and having continuously varying critical exponents, like the eight-vertex model. We also show that they are related to a large class of unitary conformal field theories.
We study the S=1S=1S=1 quantum spin chains with bilinear, biquadratic, plus bond alternation in the vicinity of the S=1S=1S=1 Takhtajan-Babujian model. Transition line between the Haldane and the dimer phases is determined numerically. To see the crossover behavior from c=3/2[k=2c=3/2 [k=2c=3/2[k=2 SU(2) Weiss-Zumino-Witten (WZW) model] at the Takhtajan-Babujian point to c=1[k=1c=1 [k=1c=1[k=1 SU(2) WZW model], we calculate the conformal anomaly c and scaling dimensions of the primary fields on the transition line.
It is known that the standard method for calculating finite-size corrections in Bethe ansatz solvable systems is not applicable to the Takhtajan-Babujian model and its anisotropic XXZ generalisation. The authors develop a new analytic method explicitly avoiding root densities and associated problems. Nonlinear integral equations are derived whose solutions yield the correct central charges c=1 and c=3/2 for the spin-1/2 and spin-1 XXZ chains, respectively. In the spin-1 case the authors obtain as a by-product the finite-size deviation of the Bethe ansatz roots from the 2-string formation.
The authors are concerned with the critical properties of the anisotropic Heisenberg chain, or XXZ model, with arbitrary integer or half-integer spin. The eigenspectra of these Hamiltonians, with periodic boundaries, are calculated for finite chains by solving numerically their associated Bethe ansatz equations. The resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified, for lattices with an even and odd number of sites. The results for spin 1 and spin 3/2 indicate that the conformal anomaly c for these models, in the gapless regime, has the value c=3S/(1+S), independent of the anisotropy, and the exponents vary continuously with the anisotropy as in the eight-vertex model. The operator content of these models indicate that the underlying field theory governing these critical spin-S models is described by composite fields formed by the product of Gaussian and Z(N) fields, with N=2S. Finally some of the irrelevant operators which produce the leading finite-size corrections of the eigenenergies are identified.