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Physical review. B, Condensed matter

Conformal invariance and critical behavior of the O(n) model on the honeycomb lattice

Murray T. Batchelor, Henk W. J. Blöte

Published Feb 1, 1989• 46 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.39.2391

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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.
The critical O(n)\mathrm{O}(n)O(n) model on a finite honeycomb lattice is solved by the Bethe-Ansatz method. Amplitudes of the dominant finite-size corrections to part of the eigenspectrum are obtained analytically. A comparison with Cardy's predictions from the theory of conformal invariance leads to the exact results for the conformal anomaly and scaling dimensions.
Finite-size scaling and transfer-matrix techniques are used to determine the conformal anomaly and critical exponents of O(n) models on the square lattice. These calculations were performed on five branches of critical points parametrised by n. The results for two of the branches agree with the known universal properties of the O(n) model as derived for the honeycomb lattice. Finite-size convergence is poor for most of the third branch, so that in this case the classification in terms of known models is uncertain. The authors' data show that the fourth branch can be interpreted in terms of a combination of a low-temperature O(n) and a critical Ising model. Finally, the numerical results for the O(n) model on the remaining branch agree accurately with the known results for the low-temperature O(n+1) model. They prove this equivalence by means of an exact mapping.
We investigate the hard-square lattice-gas model by means of transfer-matrix calculations and a finite-size-scaling analysis. Using a minimal set of assumptions we find that the spectrum of correction-to-scaling exponents is consistent with that of the exactly solved Ising model, and that the critical exponents and correlation-length amplitudes closely follow the relation predicted by conformal invariance. Assuming that these spectra are exactly identical, and conformal invariance, we determine the critical point, the conformal anomaly, and the temperature and magnetic exponents with numerical margins of 10(-11) or less. These results are in a perfect agreement with the exactly known Ising universal parameters in two dimensions. In order to obtain this degree of precision, we included system sizes as large as feasible, and used extended-precision floating-point arithmetic. The latter resource provided a substantial improvement of the analysis, despite the fact that it restricted the transfer-matrix calculations to finite sizes of at most 34 lattice units.
The Bethe-Ansatz approach is used to solve a nineteen-vertex model equivalent to a spin model with O(n) symmetry on the square lattice. The bulk free energy is determined exactly on four branches of critical points. On two of these branches, the central charge is derived from finite-size corrections. On the two other branches, the nature of the Bethe Ansatz solutions prevents us from computing the central charge analytically. In this case, our approach still allows the exact numerical determination of the free energy on strips of hundreds of sites wide.
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.