Question of universality inRPn−1andO(n)

Abstract

We argue that there is no essential violation of universality in the continuum limit of mixed RP n−1 and O(n) lattice sigma models in 2 dimensions, contrary to opposite claims in the literature.

The finite size spectrum of the 2-dimensional nonlinear σ-model

“…This effective rotator spectrum is valid in perturbation theory (PT) up to 2-loop level [22,23]. The lowest energy levels correspond to the totally symmetric tensor states, which in our notation are the J = N states (diagonal scattering or U(1) sector).…”

“…For general isospin no direct calculation exists beyond 1-loop order [22], but using the validity of the rotator spectrum up to 2-loop order [23] we haveε…”

First-order transition induced by topological defects in theO(3)principal chiral model

“…However, there are arguments against the existence of a phase transition at a finite temperature based on the σ-model approach [20][21][22] . The σ-model is the effective theory describing low-energy (weak) fluctuations, so it also describes a low-temperature behavior.…”

Pathologies of the large-N limit for RPN−1, CPN−1, QPN−1 and mixed isovector/isotensor σ-models

“…Over the past decade there has been much interest in two-dimensional lattice σ-models taking values in the real projective space RP N −1 [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16], the complex projective space CP N −1 [1,17,4,18,19,20,21,22,23,24] or the quaternionic projective space QP N −1 [25]. 1 These models are good laboratories for investigating asymptotic freedom, topological phenomena and universality; they are intermediate in complexity between two-dimensional N-vector models (σ-models taking values in the sphere S N −1 ) and four-dimensional non-abelian gauge theories.…”

“…However, for 1 < d ≤ 2 there is an Ising-like phase transition at β T = +∞, β V = β c,Ising ; and it is at present unclear whether "new" continuum limitsthat is, continuum theories whose isotensor correlation functions differ from those of the continuum limit of the pure N-vector model -can be obtained by approaching this point [11,13,14,15] (see also [48,49,50] for the analogous problem in d = 1).…”