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RG boundaries and Cardy's variational ansatz for multiple perturbations

Anatoly Konechny

Abstract

We consider perturbations of 2D CFTs by multiple relevant operators. The massive phases of such perturbations can be labeled by conformal boundary conditions. Cardy's variational ansatz approximates the vacuum state of the perturbed theory by a smeared conformal boundary state. In this paper we study the limitations and propose generalisations of this ansatz using both analytic and numerical insights based on TCSA. In particular we analyse the stability of Cardy's ansatz states with respect to boundary relevant perturbations using bulk-boundary OPE coefficients. We show that certain transitions between the massive phases arise from a pair of boundary RG flows. The RG flows start from the conformal boundary on the transition surface and end on those that lie on the two sides of it. As an example we work out the details of the phase diagram for the Ising field theory and for the tricritical Ising model perturbed by the leading thermal and magnetic fields. For the latter we find a pair of novel transition lines that correspond to pairs of RG flows. Although the mass gap remains finite at the transition lines, several one-point functions change their behaviour. We discuss how these lines fit into the standard phase diagram of the tricritical Ising model. We show that each line extends to a two-dimensional surface $ξ_{σ,c}$ in a three coupling space when we add perturbations by the subleading magnetic field. Close to this surface we locate symmetry breaking critical lines leading to the critical Ising model. Near the critical lines we find first order phase transition lines describing two-phase coexistence regions as predicted in Landau theory. The surface $ξ_{σ,c}$ is determined from the CFT data using Cardy's ansatz and its properties are checked using TCSA numerics.

RG boundaries and Cardy's variational ansatz for multiple perturbations

Abstract

We consider perturbations of 2D CFTs by multiple relevant operators. The massive phases of such perturbations can be labeled by conformal boundary conditions. Cardy's variational ansatz approximates the vacuum state of the perturbed theory by a smeared conformal boundary state. In this paper we study the limitations and propose generalisations of this ansatz using both analytic and numerical insights based on TCSA. In particular we analyse the stability of Cardy's ansatz states with respect to boundary relevant perturbations using bulk-boundary OPE coefficients. We show that certain transitions between the massive phases arise from a pair of boundary RG flows. The RG flows start from the conformal boundary on the transition surface and end on those that lie on the two sides of it. As an example we work out the details of the phase diagram for the Ising field theory and for the tricritical Ising model perturbed by the leading thermal and magnetic fields. For the latter we find a pair of novel transition lines that correspond to pairs of RG flows. Although the mass gap remains finite at the transition lines, several one-point functions change their behaviour. We discuss how these lines fit into the standard phase diagram of the tricritical Ising model. We show that each line extends to a two-dimensional surface in a three coupling space when we add perturbations by the subleading magnetic field. Close to this surface we locate symmetry breaking critical lines leading to the critical Ising model. Near the critical lines we find first order phase transition lines describing two-phase coexistence regions as predicted in Landau theory. The surface is determined from the CFT data using Cardy's ansatz and its properties are checked using TCSA numerics.
Paper Structure (23 sections, 193 equations, 49 figures, 3 tables)

This paper contains 23 sections, 193 equations, 49 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Cardy's variational ansatz
  3. Ising field theory
  4. Some generalities
  5. Pure thermal perturbation
  6. Mixed thermal and magnetic perturbations for $t<0$
  7. Mixed thermal and magnetic perturbations for $t>0$
  8. Summary
  9. Tricritical Ising model perturbed by the leading thermal and magnetic operators
  10. Landau theory
  11. Some generalities
  12. Mixed leading thermal and magnetic perturbations for $\lambda_{\epsilon}<0$
  13. Small magnetic perturbation in the presence of $\lambda_{\epsilon}>0$
  14. Phase transition in the $\lambda_{\epsilon}>0$, $\lambda_{\sigma} \ne 0$ region
  15. Switching on $\phi_{\sigma'}$ and symmetry breaking critical lines
  16. ...and 8 more sections

Figures (49)

  • Figure 1: The unstable manifold of IFT and the associated conformal boundary states.
  • Figure 2: The phase diagram of TIM perturbed by $\phi_{\sigma}$ and $\phi_{\epsilon}$. The phases are labeled by conformal boundary states together with the leading irrelevant operator defining their perturbation at finite volume.
  • Figure 3: The unstable manifold of IFT and the associated conformal boundary states.
  • Figure 4: The exact vacuum vector components $C_{i}$ against the total conformal weight of the basis vectors for $\tilde{t}=35$. The left plot represents the vacuum sector and the right plot -- the $\epsilon$-sector. The solid grey line is the exponential function $\pm\, {\cal N} e^{-{\rm weight}/(2\tilde{t})}$.
  • Figure 5: The exact vacuum vector components $C_{i}$ against the total conformal weight of the basis vectors for $\tilde{t}=5$. The left plot represents the vacuum sector and the right plot -- the $\epsilon$-sector. The solid grey line is the exponential function $\pm\, {\cal N} e^{-{\rm weight}/(2\tilde{t})}$.
  • ...and 44 more figures