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Physical Review Letters

Exact Critical Point and Critical Exponents of O(n) Models in Two Dimensions

Bernard Nienhuis

Published Oct 11, 1982• 1,010 citations
Publisher

Abstract

A two-dimensional $n$-component spin model with cubic or isotropic symmetry is mapped onto a solid-on-solid model. Subject to some plausible assumptions this leads to an analytic calculation of the critical point and some critical indices for $\ensuremath{-}2<~n<~2$.

DOI
10.1103/physrevlett.49.1062

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arXiv:cond-mat/0512369•Communications in Theoretical Physics•May 1, 2006•1 citation
A two-dimensional nnn-component spin model with cubic or isotropic symmetry is mapped onto a solid-on-solid model. Subject to some plausible assumptions this leads to an analytic calculation of the critical point and some critical indices for \ensuremath−2< n< 2\ensuremath{-}2<~n<~2\ensuremath−2< n< 2.
Critical indices for isotropic systems of nnn-dimensional spins in ($d=4\ensuremath{-}\ensuremath{\epsilon}$)-dimensional lattices are calculated to order $\ensuremath{\epsilon}$. All critical indices corresponding to perturbations of the spin probability distribution are given. Such perturbations might arise from the effects of external or crystal fields on the spin system.
The exact transition temperature, as well as some of the critical exponents, of certain three-dimensional Ising and Potts models is presented. With the analytic results supplemented by assumptions concerning the validity of universality, the critical point is identified as a Liftshitz point. On the basis of hyperscaling, the dynamic exponent zzz for two-dimensional Ising and Potts model is predicted to be given by $z\ensuremath{\nu}=2+\ensuremath{\alpha}$.
The spinor operator in the two-dimensional Ising model can be readily generalized to other self-dual models as the product of the order parameter and its dual image, the disorder operator. Recently, the exponent of this and other operators in the qqq-state Potts model received renewed attention, as the theory of conformal invariance produces complete lists of critical exponents for some of these models. In this paper we calculate the critical and tricritical indices of the spinor operator in the two-dimensional qqq-state Potts model, via a well-known mapping into a solid-on-solid model. Our results give a physical identification for the exponents predicted by the conformal theory.
The critical behavior in zero field above Tc{T}_{c}Tc​ of ferromagnets or ferroelectrics with a Hamiltonian of cubic symmetry is studied, to order ${\ensuremath{\epsilon}}^{2}$, by exact renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions with nnn-component spins. For $\ensuremath{\epsilon}=1$, $n\ensuremath{\ge}3$, a crossover from isotropic (Heisenberg) to characteristic cubic behavior occurs, with the new value $2{v}^{C}=1+[\frac{(n\ensuremath{-}1)}{3n}]\ensuremath{\epsilon}+[\frac{(n\ensuremath{-}1)}{324{n}^{3}}](17{n}^{2}+290n\ensuremath{-}424){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, and cubic symmetry appearing in the four-spin correlation function. Experiments on structural phase transitions are considered briefly.
For pt.I see J. Phys. C: Solid St. Phys., vol.6, p.2691 (1973). The critical behaviour of the step model of phase transitions on the square, simple cubic and body-centred cubic lattices is investigated by the method of exact series expansions. The model is found to be in agreement with two aspects of universality, namely the dependence of critical exponents on spin-space dimensionality and the lattice independence of critical exponents for fixed lattice and spin-space dimensionality. Series expansions for the three-dimensional susceptibility and specific heats are generated and analysed, and the critical exponents are found to be gamma =1.335+or-0.01 and alpha =-0.04+or-0.10 respectively. In two dimensions the existence and nature of a phase transition remains an open question. No evidence of either a conventional algebraic singularity or a vortex induced essential singularity is found.