Superinstantons and the Reliability of Perturbation Theory in Non-Abelian Models

Abstract

In dimension D ≤ 2 the low temperature behavior of systems enjoying a continuous symmetry is dominated by super-instantons: classical configurations of arbitrarily low energy. Perturbation theory in the background of a super-instanton produces thermodynamic answers for the invariant Green's functions that differ from the standard ones, but only in non-Abelian models and only starting at O(1/β 2 ). This effect modifies the β-function of the O(N ) models and persists in the large N limit of the O(N ) models.

The Kosterlitz-Thouless universality class

“…For earlier counterevidence to both the physical KT picture as well as the quantitative essential scaling predictions see [40,41,42] and [13,39]. Further criticisms of the conventional view are found in [43].…”

Patrascioiu and Seiler Reply:

“…In his recent Comment [1] on our Physical Review Letter [2], David accepts our computations but challenges our conclusion that in non-Abelian models perturbation theory (PT) is untrustworthy even at short distances. Moreover, he claims that if one 'renormalizes' PT appropriately, one recovers the standard answers, such as the known Callan-Symanzik β-function.…”

Some new results on an old controversy: Is perturbation theory the correct asymptotic expansion in non-Abelian models?

“…We have nothing to contribute to (ii), but some new computations confirming (i) beyond what was stated in [2]. All our computations, just as those of [1,2] refer to a special obervable, namely the nearest neighbor spin-spin scalar product located in the center of the lattice, i.e. O = s(0, 0) · s(0, 1).…”

“…Several years ago A. Patrascioiu and the second-named author [1] investigated the dependence of perturbation theory (PT) on boundary conditions (b.c.). Surprisingly they found that in two-dimensional nonlinear sigmamodels with non-abelian symmetry the coefficients of the perturbative expansion depend on the boundary conditions (b.c.)…”