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On the renormalisation group for the boundary Truncated Conformal Space Approach

Gerard Watts

TL;DR

<3-5 sentence high-level summary> The paper analyzes the truncated conformal space approach (TCSA) for perturbed boundary conformal field theories, identifying two leading truncation effects: renormalisation of the coupling constants and a multiplicative renormalisation of the energy scale. It derives these effects from physical (operator product expansions) and mathematical (eigenvalue analysis) arguments, predicting scaling forms for the renormalisation function $g_n$ and the energy rescaling $r_n$, and validates them with detailed checks in the tri-critical Ising and Yang-Lee models. It also studies irrelevant (non-renormalisable) perturbations, showing that the renormalised and rescaled spectrum still matches exact results and reveals a relationship between IR and UV couplings; further, the large-coupling regime captures several fixed points and extended RG flows. Overall, the work provides a quantitative RG interpretation of TCSA errors and demonstrates predictive power of TCSA across multiple boundary flows, including flows beyond fixed points.

Abstract

In this paper we continue the study of the truncated conformal space approach to perturbed boundary conformal field theories. This approach to perturbation theory suffers from a renormalisation of the coupling constant and a multiplicative renormalisation of the Hamiltonian. We show how these two effects can be predicted by both physical and mathematical arguments and prove that they are correct to leading order for all states in the TCSA system. We check these results using the TCSA applied to the tri-critical Ising model and the Yang-Lee model. We also study the TCSA of an irrelevant (non-renormalisable) perturbation and find that, while the convergence of the coupling constant and energy scales are problematic, the renormalised and rescaled spectrum remain a very good fit to the exact result, and we find a numerical relationship between the IR and UV couplings describing a particular flow. Finally we study the large coupling behaviour of TCSA and show that it accurately encompasses several different fixed points.

On the renormalisation group for the boundary Truncated Conformal Space Approach

TL;DR

<3-5 sentence high-level summary> The paper analyzes the truncated conformal space approach (TCSA) for perturbed boundary conformal field theories, identifying two leading truncation effects: renormalisation of the coupling constants and a multiplicative renormalisation of the energy scale. It derives these effects from physical (operator product expansions) and mathematical (eigenvalue analysis) arguments, predicting scaling forms for the renormalisation function and the energy rescaling , and validates them with detailed checks in the tri-critical Ising and Yang-Lee models. It also studies irrelevant (non-renormalisable) perturbations, showing that the renormalised and rescaled spectrum still matches exact results and reveals a relationship between IR and UV couplings; further, the large-coupling regime captures several fixed points and extended RG flows. Overall, the work provides a quantitative RG interpretation of TCSA errors and demonstrates predictive power of TCSA across multiple boundary flows, including flows beyond fixed points.

Abstract

In this paper we continue the study of the truncated conformal space approach to perturbed boundary conformal field theories. This approach to perturbation theory suffers from a renormalisation of the coupling constant and a multiplicative renormalisation of the Hamiltonian. We show how these two effects can be predicted by both physical and mathematical arguments and prove that they are correct to leading order for all states in the TCSA system. We check these results using the TCSA applied to the tri-critical Ising model and the Yang-Lee model. We also study the TCSA of an irrelevant (non-renormalisable) perturbation and find that, while the convergence of the coupling constant and energy scales are problematic, the renormalised and rescaled spectrum remain a very good fit to the exact result, and we find a numerical relationship between the IR and UV couplings describing a particular flow. Finally we study the large coupling behaviour of TCSA and show that it accurately encompasses several different fixed points.

Paper Structure

This paper contains 21 sections, 85 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The TCSA approach and its errors
  3. The TCSA approach
  4. The tri-critical Ising model numerical results
  5. Physical Arguments
  6. The TCSA renormalisation group
  7. The coupling constant renormalisation
  8. The energy rescaling
  9. Corrections from the identity operator
  10. Mathematical Arguments
  11. Checks in the tri-critical Ising model
  12. The coupling constant renormalisation
  13. The energy rescaling
  14. Checks in the Yang-Lee model
  15. Fixed points and the coupling constant renormalisation.
  16. ...and 6 more sections

Figures (11)

  • Figure 1: The energy gaps for the strip with boundary conditions $(11)$ and $(12) + \lambda\phi_{(13)}$ as given by the TBA and TCSA methods plotted against $\lambda$.
  • Figure 2: The numerical coupling constant renormalisation and energy rescaling found for the tri-critical Ising model at truncation levels 6 (red, solid), 14 (green, dotted) and 22 (blue, dashed).
  • Figure 3: The space of boundary flows in the tricritical Ising model
  • Figure 4: The numerical coupling constant renormalisation found for the tri-critical Ising model with $(11)$ boundary condition on the other edge, for truncation levels 8 ($\circ$) 15 ($\bullet$) and 22 ($\square$) together with the 1-loop predictions (\ref{['eq:gn1loop']}) shown as a solid line.
  • Figure 5: The coupling constant and energy rescaling functions
  • ...and 6 more figures