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Journal of Physics A Mathematical and General

Exact results for the critical behaviour of a Nienhuis O(n) model on the square lattice

Murray T. Batchelor

Published Aug 7, 1993• 9 citations
Publisher

Abstract

Exact results are obtained for a square lattice fifteen-vertex model and related O(n) model recently proposed by Nienhuis (1990). The partition function per site and finite-size corrections are obtained, along with the critical exponents via the Temperley-Lieb equivalent six-vertex model. The exact exponents nu = 4 / 7 and gamma = 6 / 7 follow in the limit n=0 for even system sizes. Odd system sizes are also discussed. The fifteen-vertex model is also the vertex formulation of the anisotropic, or q-deformed, spin-1 biquadratic model.

DOI
10.1088/0305-4470/26/15/023

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Physical review. B, Condensed matter•Feb 1, 1989•46 citations
Exact results are obtained for a square lattice fifteen-vertex model and related O(n) model recently proposed by Nienhuis (1990). The partition function per site and finite-size corrections are obtained, along with the critical exponents via the Temperley-Lieb equivalent six-vertex model. The exact exponents nu = 4 / 7 and gamma = 6 / 7 follow in the limit n=0 for even system sizes. Odd system sizes are also discussed. The fifteen-vertex model is also the vertex formulation of the anisotropic, or q-deformed, spin-1 biquadratic model.
We find and discuss the scaling dimensions of the branch 0 manifold of the Nienhuis O(n) loop model on the square lattice, concentrating on the surface dimensions. The results are extracted from a Bethe ansatz calculation of the finite-size corrections to the eigenspectrum of the six-vertex model with free boundary conditions. These results are especially interesting for polymer physics at two values of the crossing parameter lambda . Interacting self-avoiding walks on the Manhattan lattice at the collapse temperature ( lambda = pi /3) and Hamiltonian walks on the Manhattan lattice ( lambda = pi /2) are discussed in detail. Our calculations illustrate the importance of examining both odd and even strip widths when performing finite-size correction calculations to obtain scaling dimensions.
A study is made of the anisotropic biquadratic spin-1 Hamiltonian introduced by Batchelor et al. (1990). This model can be represented as a sum of operators satisfying a Temperley-Lieb algebra. Its critical behaviour is explored by using the relations between mass-gap amplitudes and critical exponents predicted by conformal invariance. The model, like in the XXZ chain, is found to exhibit a line of continuously varying exponents. By comparing its eigenspectrum with that of the XXZ Hamiltonian, several exact values are predicted.
The partition function of the symmetric (zero electric field) eight-vertex model on a square lattice can be formulated either in the original "electric" vertex format or in an equivalent "magnetic" Ising-spin format. In this paper, both electric and magnetic versions of the model are studied numerically by using the corner transfer matrix renormalization-group method which provides reliable data. The emphasis is put on the calculation of four specific critical exponents, related by two scaling relations, and of the central charge. The numerical method is first tested in the magnetic format, the obtained dependencies of critical exponents on the model's parameters agree with Baxter's exact solution, and weak universality is confirmed within the accuracy of the method due to the finite size of the system. In particular, the critical exponents η and δ are constant as required by weak universality. On the other hand, in the electric format, analytic formulas based on the scaling relations are derived for the critical exponents η_{e} and δ_{e} which agree with our numerical data. These exponents depend on the model's parameters which is evidence for the full nonuniversality of the symmetric eight-vertex model in the original electric formulation.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems of square geometry with periodic boundaries oriented either along <10> or along <11> direction have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction proportional to 1/L^0.25 of a very small magnitude also has been detected in agreement with predictions of our recently developed GFD (grouping of Feynman diagrams) theory. A refined analysis of the recent MC data for 3D Ising, phi^4, and XY lattice models has been performed. It includes an analysis of the partition function zeros of 3D Ising model, an estimation of the correction-to-scaling exponent omega from the Binder cumulant data near criticality, as well as a study of the effective critical exponent eta and the effective amplitudes in the asymptotic expansion of susceptibility at the critical point. In all cases a refined analysis is consistent with our (GFD) asymptotic values of the critical exponents (nu=2/3, omega=1/2, eta=1/8 for 3D Ising model and omega=5/9 for 3D XY model), while the actually accepted "conventional" exponents are, in fact, effective exponents which are valid for approximation of the finite--size scaling behavior of not too large systems.
Abstract We compute the exact partition function for the 3-state Potts model on square lattices of several sizes larger than previously accessible. Making comparison with the exactly solved Ising model we show that, for aspects of the analytic structure close to the ferromagnetic transition point, these lattices are large enough to approach the thermodynamic limit. Subject to certain assumptions this allows for computation of estimates for the specific heat critical exponent. We thus obtain an estimate for this exponent. The estimate is consistent with the known result, thus demonstrating the potential use of this method for other models. We also discuss the antiferromagnetic transition.
A seven-vertex model on the honeycomb lattice is solved exactly by the Bethe ansatz method. The vertex model is equivalent to the critical O(n) model on the honeycomb lattice. The equivalence is made exact for lattices wrapped on a cylinder with one finite and one infinite direction, by the introduction of a seam into the vertex model. Thus asymptotic finite-size amplitudes of the critical O(n) model are obtained exactly. Applications of the theory of conformal invariance, which relate these amplitudes to the central charge and critical exponents, confirm the existing results and conjectures for these quantities. The finite-size amplitude corresponding to the temperature exponent was not obtained from the exact solution. However, this amplitude was accurately determined from numerical finite-size results obtained by the transfer-matrix method. This result also agrees with recent predictions.