Study of the asymptotic freedom of 2D Yukawa models on the lattice

Abstract

We investigate on the lattice the Yukawa models in 2 dimensions with Z(2) and U(1) symmetries. These models reduce to the usual and chiral Gross-Neveu models, respectively, when the kinetic and the selfcoupling terms of the scalar field are turned off. The numerical data and mean field arguments suggest that, at least for some range of the scalar field hopping parameter, fermion mass is dynamically generated for arbitrarily weak Yukawa coupling. The models are asymptotically free in this coupling, like the Gross-Neveu models, even when the scalar quartic selfcoupling is strong.

On the equivalence between 2D Yukawa and Gross-Neveu models

“…As discussed in [10] the MF approximation leads also to a better understanding of the fermion mass generation mechanism in the case of the Y 2 model with continuous chiral symmetry which cannot be broken spontaneously according to the Mermin-Wagner-Coleman theorem. The main idea is that the effective interaction induced by the fermions can produce ferromagnetic ordering of scalar variables on distances O(1/m F ).…”

“…This property is suggested by the MF method [10] and in the effective potential approach [12] which is in spirit similar to the MF approximation. It is argued in ref.…”

“…In [10] we argued that the MF approximation can be used to describe the scaling behavior and also some finite cutoff and finite size effects in the Y 2 models for any λ. The fermion determinant generates nonlocal ferromagnetic interactions between φ's.…”

“…It is argued in ref. [10] that the MF method is applicable to the Y 2 models because the fermions induce long range effective interaction which couples many scalar variables ferromagnetically. Such interaction can be well described by a mean field.…”

“…The agreement between the numerical and MF methods obtained until now suggests to test numerically for various values of the coupling parameters and with improved data the MF inspired conjecture [8,12,10] that the Y 2 and the GN 2 models with the same symmetry belong to the same universality class, i.e. correspond to the same renormalized quantum field theory and have the same physical content in a broad range of the coupling parameters.…”