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arXiv:hep-lat/9508014 · Physical review. B, Condensed matter

Universality class ofO(N) models

Adrian Patrascioiu, Erhard Seiler

Published Sep 1, 1996• 31 citations
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Abstract

We point out that existing numerical data on the correlation length and magnetic susceptibility suggest that the two-dimensional O(3) model with standard action has a critical exponent \ensuremath{\eta}=1/4, which is inconsistent with asymptotic freedom. This value of \ensuremath{\eta} is also different from the one of the Wess-Zumino-Novikov-Witten model that is supposed to correspond to the O(3) model at \ensuremath{\theta}=\ensuremath{\pi}. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.7177

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We point out that existing numerical data on the correlation length and magnetic susceptibility suggest that the two-dimensional O(3) model with standard action has a critical exponent \ensuremath{\eta}=1/4, which is inconsistent with asymptotic freedom. This value of \ensuremath{\eta} is also different from the one of the Wess-Zumino-Novikov-Witten model that is supposed to correspond to the O(3) model at \ensuremath{\theta}=\ensuremath{\pi}. \textcopyright{} 1996 The American Physical Society.
In this paper, I revise arguments in favor of the PSL(2\ensuremath{\mid}2) Wess-Zumino-Novikov-Witten (WZNW) model as a theory of the plateau transition in integer quantum Hall effect. I show that all available numerical data (including the correlation length exponent $\ensuremath{\nu}$) are consistent with the predictions of such WZNW model with the level k=8k=8k=8.
We study the critical behavior at the ordinary surface universality class of the three-dimensional O(N)\mathrm{O}(N)O(N) model, bounded by a two-dimensional surface. Using high-precision Monte Carlo simulations of an improved lattice model, where the leading bulk scaling correction is suppressed, and finite-size scaling analysis of the fourth cumulant of the surface magnetization, we obtain precise estimates of the scaling dimension of the surface field operator for N=2,3,4N=2,3,4N=2,3,4. We also determine the fixed-point values of two renormalization-group invariant observables, which characterize the finite-size scaling behavior at the ordinary transition.
It is known that the U ( 2 ) Wess-Zumino-Witten model is dual to the free fermion theory in two dimensions via non-Abelian bosonization. While it is decomposed into the S U ( 2 ) Wess-Zumino-Witten model and a free compact boson, the former is believed to be equivalent to the O ( 3 ) nonlinear sigma model with the theta term at θ = π . In this work, we reexamine this duality through the lens of nonperturbative renormalization group (RG) flow. We analyze the RG flow structure of the O ( 3 ) nonlinear sigma model with the theta term in two dimensions using the functional renormalization group. Our results reveal a nontrivial fixed point with a nonzero value of the topological coupling. The scaling dimensions (critical exponents) at this fixed point suggest the realization of a duality between the O ( 3 ) nonlinear sigma model with the theta term and the free fermion theory, indicating that these models belong to the same universality class.