Renormalization group flows for the second $Z_{N}$ parafermionic field theory for N odd

Vladimir S. Dotsenko, Benoit Estienne

Published Jan 1, 2007• 0 citations

Abstract

Using the renormalization group approach, the Coulomb gas and the coset techniques, the effect of slightly relevant perturbations is studied for the second parafermionic field theory with the symmetry $Z_{N}$, for N odd. New fixed points are found and classified.

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Renormalization group flows for the second $Z_{N}$ parafermionic field theory for N oddRenormalization group flows for the second $Z_{N}$ parafermionic field theory for N odd

Vladimir S. Dotsenko, Benoit Estienne

Published Jan 1, 2007• 0 citations

OpenAlex

Abstract

Z_{N}

, for N odd. New fixed points are found and classified.

Related papers

A lattice model for the second $\mathbb{Z}_{3}$ parafermionic field theoryA lattice model for the second $\mathbb{Z}_{3}$ parafermionic field theory

Benoit Estienne

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Renormalization group flows for the second

Z_{N}

parafermionic field theory for N odd

Z_{N}

, for N odd. New fixed points are found and classified.

A lattice model for the second

\mathbb{Z}_{3}

parafermionic field theory

The second

\mathbb{Z}_{3}

parafermionic conformal theories are associated with the coset construction $\frac{SU(2)_{k}\times SU(2)_{4}}{SU(2)_{k+4}} $. Solid-on-solid integrable lattice models obtained by fusion of the model based on level-1 representation of the affine algebra

B_1^{(1)}

have a critical point described by these conformal theories. Explicit values for the Boltzmann weights are derived for these models, and it is shown that the Boltzmann weights can be made positive for a particular value of the spectral parameter, opening a way to eventual numerical simulations of these conformal field theories. Away from criticality, these lattice models describe an integrable, massive perturbation of the parafermionic conformal theory by the relevant field $Ψ_{-2/3}^{\dagger}D_{1,3} $.

{\mathbb{Z}}_{2}

-symmetric critical point which are characterized by an upper critical dimension ${d}_{c}=2nk/(n\ensuremath{-}1)$ accumulating at even integers. We distinguish two types of theories depending on whether or not the numbers

k

and $n\ensuremath{-}1$ are relatively prime. When they are, the critical theory involves a marginal powerlike interaction ${\ensuremath{\phi}}^{2n}$ and the deformations admit a derivative expansion that at leading order involves only the potential. In this case we present the beta functional of the potential and use this to calculate some anomalous dimensions and operator product expansion coefficients. These confirm some conformal field theory data obtained using conformal-block techniques, while giving new results. In the second case where

k

and $n\ensuremath{-}1$ have a common divisor, the theories show a much richer structure induced by the presence of marginal derivative operators at criticality. We study the case

k=2

with odd values of

n

, which fall in the second class, and calculate the functional flows and spectrum. These theories have a phase diagram characterized at leading order in $\ensuremath{\epsilon}$ by four fixed points which apart from the Gaussian UV fixed point include an IR fixed point with a purely derivative interaction.

We consider a RG flow in a general

\hat{su}(2)

coset model perturbed by the least relevant field. The perturbing field as well as some particular fields of dimension close to one are constructed recursively in terms of lower level fields. Using this construction we obtain the structure constants and the four-point correlation functions in the leading order. This allows us to compute the mixing coefficients among the fields in the UV and the IR theory.