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Physical Review Letters

Conformal Anomaly and Scaling Dimensions of the O(n) Model from an Exact Solution on the Honeycomb Lattice

Murray T. Batchelor, Henk W. J. Blöte

Published Aug 22, 1988• 2 citations
Publisher

Abstract

dependently by two other groups.The first observation was done by Glorieux and co-workers, ' and Hennequin
DOI
10.1103/physrevlett.61.1042.3

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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.
We derive the exact critical couplings (x∗,ya∗)({x}^{*}{,y}_{a}^{*})(x∗,ya∗​), where ya∗/x∗=1+2=1.533…{y}_{a}^{*}{/x}^{*}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\sqrt{1+\sqrt{2}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.533\dots{}ya∗​/x∗=1+2​​=1.533… , for the polymer adsorption transition on the honeycomb lattice, along with the universal critical exponents, from the Bethe ansatz solution of the O(n)O\left(n\right)O(n) loop model at the special transition. Our result for the thermal scaling dimension, and thus the crossover exponent $\ensuremath{\varphi}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$, is in agreement with an earlier result based on conformal invariance arguments. Our result for the geometric scaling dimensions confirms recent conjectures that they are given by ${h}_{\ensuremath{\ell}+1,3}$ in the Kac formula.
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.