Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Criticality in self-dual sine-Gordon models

P. Lecheminant, A. O. Gogolin, A. A. Nersesyan

TL;DR

This work addresses criticality in the self-dual extension of the sine-Gordon model at $\beta^2 = 2\pi N$, focusing on the non-perturbative $N=3$ case where a $\mathbb{Z}_3$ (three-state Potts) critical point is predicted. It develops two independent non-perturbative routes to the IR fixed point for $\beta^2 = 6\pi$: (i) an exact mapping to an integrable deformation of the $\mathbb{Z}_4$ parafermion CFT that exhibits a massless flow to the $\mathcal{M}_5$ (Potts) fixed point, and (ii) a relationship to a chirally asymmetric $\mathrm{su}(2)_4 \otimes \mathrm{su}(2)_1$ WZW model with anisotropic current-current interactions leading to a Toulouse decoupling point and the same $\mathbb{Z}_3$ IR physics. The paper also develops a complementary chirally stabilized liquids perspective, showing how the IR Potts criticality emerges from a massless flow in the anisotropic WZW framework, and discusses UV–IR transmutations of operators. These results solidify the connection between SDSG criticality and $\mathbb{Z}_3$ universality, with implications for two-dimensional XY systems with $N$-fold symmetry breaking and for one-dimensional quantum spin/electron systems.

Abstract

We discuss the nature of criticality in the $β^2 = 2 πN$ self-dual extention of the sine-Gordon model. This field theory is related to the two-dimensional classical XY model with a N-fold degenerate symmetry-breaking field. We briefly overview the already studied cases $N=2,4$ and analyze in detail the case N=3 where a single phase transition in the three-state Potts universality class is expected to occur. The Z$_3$ infrared critical properties of the $β^2 = 6 π$ self-dual sine-Gordon model are derived using two non-perturbative approaches. On one hand, we map the model onto an integrable deformation of the Z$_4$ parafermion theory. The latter is known to flow to a massless Z$_3$ infrared fixed point. Another route is based on the connection with a chirally asymmetric, su(2)$_4$ $\otimes$ su(2)$_1$ Wess-Zumino-Novikov-Witten model with anisotropic current-current interaction, where we explore the existence of a decoupling (Toulouse) point.

Criticality in self-dual sine-Gordon models

TL;DR

This work addresses criticality in the self-dual extension of the sine-Gordon model at , focusing on the non-perturbative case where a (three-state Potts) critical point is predicted. It develops two independent non-perturbative routes to the IR fixed point for : (i) an exact mapping to an integrable deformation of the parafermion CFT that exhibits a massless flow to the (Potts) fixed point, and (ii) a relationship to a chirally asymmetric WZW model with anisotropic current-current interactions leading to a Toulouse decoupling point and the same IR physics. The paper also develops a complementary chirally stabilized liquids perspective, showing how the IR Potts criticality emerges from a massless flow in the anisotropic WZW framework, and discusses UV–IR transmutations of operators. These results solidify the connection between SDSG criticality and universality, with implications for two-dimensional XY systems with -fold symmetry breaking and for one-dimensional quantum spin/electron systems.

Abstract

We discuss the nature of criticality in the self-dual extention of the sine-Gordon model. This field theory is related to the two-dimensional classical XY model with a N-fold degenerate symmetry-breaking field. We briefly overview the already studied cases and analyze in detail the case N=3 where a single phase transition in the three-state Potts universality class is expected to occur. The Z infrared critical properties of the self-dual sine-Gordon model are derived using two non-perturbative approaches. On one hand, we map the model onto an integrable deformation of the Z parafermion theory. The latter is known to flow to a massless Z infrared fixed point. Another route is based on the connection with a chirally asymmetric, su(2) su(2) Wess-Zumino-Novikov-Witten model with anisotropic current-current interaction, where we explore the existence of a decoupling (Toulouse) point.

Paper Structure

This paper contains 15 sections, 62 equations.

Table of Contents

  1. Introduction
  2. Review of some simple cases
  3. The $\beta^2 = 2\pi$ SDSG model
  4. The $\beta^2 = 4\pi$ SDSG model
  5. The $\beta^2 = 8\pi$ SDSG model
  6. Mapping of the $\beta^2=6\pi$ SDSG model onto an integrable deformation of the Z$_4$ parafermion theory
  7. The $\beta^2 = 6 \pi$ SDSG model and chirally stabilized liquids
  8. Chirally stabilized liquids
  9. Decoupling bosonic degrees of freedom
  10. Three-state Potts criticality
  11. The UV-IR transmutation of the fields
  12. Concluding remarks
  13. Bosonization of the Z$_4$ parafermion theory
  14. Identification of the Bose field
  15. Bosonization of the Z$_4$ parafermionic currents