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Physical Review Letters

Critical Spin Models with Continuously Varying Exponents

Francisco Castilho Alcaraz, Marcio Jose Martins

Published Oct 3, 1988• 33 citations
Publisher

Abstract

The eigenenergies for the spin $S$ generalization of the critical $\mathrm{XXZ}$ 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 $c=\frac{3S}{(S+1)}$ 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.

DOI
10.1103/physrevlett.61.1529

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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.
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The eigenspectrum of the critical anisotropic Heisenberg model (or XXZ model), with arbitrary spin S in its disordered-ferroelectric regime (0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}\ensuremath{\pi}/2S) is solved by the Bethe-Ansatz method. The amplitudes of the leading finite-size corrections are calculated analytically and numerically. Using conformal invariance we give exact results for the conformal anomaly and scaling dimensions. Our results indicate that, for all spin S, the critical behavior is governed by a conformal field theory with central charge c=1.
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.
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Eigenspectra of the critical quantum Ashkin-Teller and Potts chains with free boundaries can be obtained from that of the XXZ chain with free boundaries and a complex surface field. By deriving and solving numerically the Bethe ansatz equations for such boundaries the authors obtain eigenenergies of XXZ chains of up to 512 sites. The conformal anomaly and surface exponents of the quantum XXZ, Ashkin-Teller, and Potts chains are calculated by exploiting their relations with the mass gap amplitudes as predicted by conformal invariance.
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