Functional Renormalization Group Approach to the Sine-Gordon Model

Abstract

The renormalization group flow is presented for the two-dimensional sine-Gordon model within the framework of the functional renormalization group method by including the wave-function renormalization constant. The Kosterlitz-Thouless-Berezinski type phase structure is recovered as the interpolating scaling law between two competing IR attractive area of the global renormalization group flow.

Renormalization of QCD2

“…However the bosonized version of the toy models, which are simple scalar models can be easily treated by the functional RG method. The low energy IR physics of the 2-dimensional one-component scalar field theories which contain periodic self-interaction term are well understood [14,17,31,32].…”

“…The functional RG method showed [31,32,56,57] that this SG model has two phases depending on the value of its parameter β 2 /3. As we have shown previously in section 3.1 , the multi-color QCD 2 gives values of the parameter β 2 = 4π < β 2 c (N ) for arbitrary number N > 1 of colors, which means that the corresponding SG model is in the symmetry broken phase.…”

“…The latter case corresponds to the sharp cutoff limit, which makes the functional form of the CS and the sharp cutoff (or Wegner-Houghton type) evolution equations similar. The effective action is expanded in powers of the derivative of the field, [32,48] but gives slight modifications around β 2 = 4π [17], therefore it is not supposed to affect the phase structure of the model. Thus we do not go beyond the LPA, and set Z k [Φ] = 1.…”

“…It implies that the investigation which is confined to the local potential approximation (LPA) where β does not evolve may give reliable evolutions. Although the Wilsonian renormalization procedure is very powerful in LPA [33], we choose the effective average action RG method [32,[34][35][36][37][38][39][40][41]] to obtain the evolution of the models due to its more flexible usage. The RG evolution provides us the scale dependence of the couplings in the scalar model from which the scaling of the original fermionic couplings can be obtained according to the bosonization rules.…”

“…The systematic removal of the quantum fluctuations can be performed by the evolution equation for the effective action [32,[34][35][36][37][38][39][40][41] …”