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Lifshitz critical points meet Zamolodchikov perturbation theory

António Antunes

Abstract

Critical points of classical and quantum lattice models are often described by scale-invariant Lifshitz theories which are anisotropic in the continuum limit, as characterized by a dynamical critical exponent $z\neq1$. This type of critical behavior can in principle be studied by deforming ordinary $z=1$ conformal field theories (CFTs) by relevant vector operators breaking the rotational/Lorentz symmetry. In this short note, we consider a two-dimensional system of coupled minimal model CFTs $\mathcal{M}_{m,m+1}$ which realizes this perspective in a controlled fashion via Zamolodchikov's large $m$ expansion. The model turns out to exhibit interesting properties, including a manifold of interacting Lifshitz fixed points and emergent rotational symmetry in the infrared.

Lifshitz critical points meet Zamolodchikov perturbation theory

Abstract

Critical points of classical and quantum lattice models are often described by scale-invariant Lifshitz theories which are anisotropic in the continuum limit, as characterized by a dynamical critical exponent . This type of critical behavior can in principle be studied by deforming ordinary conformal field theories (CFTs) by relevant vector operators breaking the rotational/Lorentz symmetry. In this short note, we consider a two-dimensional system of coupled minimal model CFTs which realizes this perspective in a controlled fashion via Zamolodchikov's large expansion. The model turns out to exhibit interesting properties, including a manifold of interacting Lifshitz fixed points and emergent rotational symmetry in the infrared.
Paper Structure (11 sections, 63 equations, 1 figure)

This paper contains 11 sections, 63 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Brief review of Lifshitz theory
  3. Basic properties
  4. Example models
  5. Cardy's approach to the chiral Potts model
  6. Conformal perturbation theory with vector deformations
  7. The Lifshitz-Zamolodchikov model
  8. Zamolodchikov's perturbation theory
  9. Coupling minimal models
  10. Conclusions and Outlook
  11. Acknowledgements

Figures (1)

  • Figure 1: RG flow diagram for the Lifshitz-Zamolodchikov model \ref{['LifZam']}. The UV fixed point is denoted in blue and the rotationally invariant IR fixed point is denoted in red. Generic flows connect these two points and are denoted by the black arrows. The circle of RG unstable Lifshitz fixed points is denoted in orange and the fine-tuned flows which hit these points are denoted by the dashed lines.