Defect flows in minimal models

TL;DR

This work analyzes defect RG flows in Virasoro minimal models $M(p,p{+}1)$ using a blend of exact results, perturbative RG, and the truncated conformal space approach (TCSA). It reveals a two-parameter defect-flow space generated by perturbations of a topological $(1,2)$ defect and identifies six IR fixed points near $p>3$: two are topological, one is a sum of factorising defects, and one is a novel non-topological conformal defect $C$, with the remaining consistent with an identity-type endpoint; in the Ising case ($p=3$) the results align with Fendley et al. The analysis employs folding to relate defects to boundary conditions, classifies defect types (topological, factorising, and conformal), and uses TCSA to confirm fixed points and probe spectra on strips. The findings provide a structured picture of defect flows, including a non-trivial conformal defect at large $p$, and connect perturbative predictions with numerical spectra, offering a framework for exploring defect spaces in rational CFTs. The results have implications for understanding defect-induced RG flows, defect entropy via $g$-values, and the role of folded-model descriptions in identifying conformal defects.

Abstract

In this paper we study a simple example of a two-parameter space of renormalisation group flows of defects in Virasoro minimal models. We use a combination of exact results, perturbation theory and the truncated conformal space approach to search for fixed points and investigate their nature. For the Ising model, we confirm the recent results of Fendley et al. In the case of central charge close to one, we find six fixed points, five of which we can identify in terms of known defects and one of which we conjecture is a new non-trivial conformal defect. We also include several new results on exact properties of perturbed defects and on the renormalisation group in the truncated conformal space approach.

Figures (13)

• Figure 1: The proposed flows for the perturbation \ref{['eq:MM-12-pert']} for $p>3$. The point $D$ is the $(1,2)$-defect. The possible endpoints are: $I$ -- the identity defect, $D'$ -- the $(2,1)$-defect, $F$ -- a factorising defect given by the sum $\sum_{r=1}^{p-1} {\vert\space\vert\space r,1\space\rangle\!\rangle} {\langle\!\langle\space r,1\space\vert\space\vert}$ of $p{-}1$ conformal boundary conditions, and finally $C$ -- the new conformal defect. For details see the body of the paper.
• Figure 2: The equivalence between a defect in a CFT and a boundary condition in the folded model
• Figure 3: The equivalence between a factorising defect in a CFT and separate boundary conditions in the folded model
• Figure 4: The Dirichlet (top) and Neumann (bottom) boundary conditions on the orbifolded free boson and the corresponding topological and factorising defects in the folded Ising model
• Figure 5: The perturbative flows for the system \ref{['eq:pflows']}. The perturbative fixed points are as in figure \ref{['fig:flow-p>3']}, the black diamond being the conformal defect $C$.