Nontrivial fixed point in nonabelian models

We investigate the percolation properties of equatorial strips in the two-dimensional O(3) nonlinear $\sigma$ model. We find convincing evidence that such strips do not percolate at low temperatures, provided they are sufficiently narrow. Rigorous arguments show that this implies the vanishing of the mass gap at low temperature and the absence of asymptotic freedom in the massive continuum limit. We also give an intuitive explanation of the transition to a massless phase and, based on it, an estimate of the transition temperature.Comment: Lattice 2000 (Perturbation Theory

Does Conformal Quantum Field Theory Describe the Continuum Limits of 2D Spin Models with Continuous Symmetry?

It is generally taken for granted that two-dimensional critical phenomena can be fully classified by the well known two-dimensional (rational) conformal quantum field theories (CQFTs). In particular it is believed that in models with a continuous symmetry characterized by a Lie group $G$ the continuum theory enjoys an enhanced symmetry $G\times G$ due to the decoupling of right and left movers. In this letter we review the conventional arguments leading to this conclusion, point out two gaps and provide a conterexample. Nevertheless we justify in the end the conventional conclusions by additional arguments.Comment: 9 page

Testing Asymptotic Scaling and Nonabelian Symmetry Enhancement

We determine some points on the finite size scaling curve for the correlation length in the two dimensional O(3) and icosahedron spin models. The Monte Carlo data are consistent with the two models possessing the same continuum limit. The data also suggest that the continuum scaling curve lies above the estimate of Kim and of Caracciolo et al and thus leads to larger thermodynamic values of of the correlation length than previously reported.Comment: 10 pages, 3 figures. Some typos corrected and a discussion of recent work by Caracciolo et al include

Is the 2D O(3) Nonlinear σ\sigma Model Asymptotically Free?

We report the results of a Monte Carlo study of the continuum limit of the two dimensional O(3) non-linear $\sigma$ model. The notable finding is that it agrees very well with both the prediction inspired by Zamolodchikovs' S-matrix ansatz and with the continuum limit of the dodecahedron spin model. The latter finding renders the existence of asymptotic freedom in the O(3) model rather unlikely.Comment: 10 page

Questionable Arguments for the Correctness of Perturbation Theory in Non-Abelian Models

We analyze the arguments put forward recently by Niedermayer et al in favor of the correctness of conventional perturbation theory in non-Abelian models and supposedly showing that our super-instanton counterexample was sick. We point out that within their own set of assumptions, the proof of Niedermayer et al regarding the correctness of perturbation theory is incorrect and provide a correct proof under more restrictive assumptions. We reply also to their claim that the S-matrix bootstrap approach of Balog et al supports the existence of asymptotic freedom in the O(3) model.Comment: 9 page

Nonlinear σ\sigma-model, form factors and universality

We report the results of a very high statistics Monte Carlo study of the continuum limit of the two dimensional O(3) non-linear $\sigma$ model. We find a significant discrepancy between the continuum extrapolation of our data and the form factor prediction of Balog and Niedermaier, inspired by the Zamolodchikovs' S-matrix ansatz. On the other hand our results for the O(3) and the dodecahedron model are consistent with our earlier finding that the two models possess the same continuum limit.Comment: 10 pages, 5 figure