Block-spin renormalization group improved non-linear σ model

The β function of the two-dimensional non-linear σ model

“…The Symanzik improvement program is based on a systematic perturbative expansion to cancel the lattice artifact corrections to scaling [2]. The idea behind the real space renormalization group improvement method is to find the renormalized trajectory of a given block transformation and to perform a Monte Carlo simulation along it [3][4][5]. These improved actions showed earlier asymptotic scaling behavior, however, the question arises: Is the behavior of the standard action model at correlation length ~--2-10 a real non-scaling behavior or is it a scaling behavior in the general sense: scaling according to the full/~ function?…”

Four loop results for the 2D O(n) nonlinear σ model with 0-loop and 1-loop Symanzik actions

“…This difficulty occurs again in the 2D nonlinear σ-model. For instance, the calculation of the mass gap in this model with O(3) symmetry has yielded results [7][8][9][10][11][12][13][14][15] in conflict with the exact analytical value m HMN of Ref. [16].…”

Four-loop free energy for the 2D O(n) non-linear σ-model with 0-loop and 1-loop Symanzik improved actions

“…For instance, the calculation of the mass gap in this model with O(3) symmetry has yielded results [7][8][9][10][11][12][13][14][15] in conflict with the exact analytical value m HMN of Ref. [16].…”