New families of flows between two-dimensional conformal field theories

TL;DR

The paper develops a massless nonlinear integral equation to describe renormalisation group flows between two-dimensional conformal field theories, focusing on nonunitary minimal models perturbed by φ21 and φ15. By analyzing the finite-volume effective central charge c_eff(r) across scales, the authors predict infinite families of flows labeled by an index I, with monotone c_eff for |I|=1 and nonmonotonic, oscillatory behavior for |I|>1. Perturbative checks in the UV and IR demonstrate consistency with conformal perturbation theory and CPT coefficients, and comparisons with known massless TBA results validate the approach where available. The work reveals a rich, nonmonotonic landscape of RG flows and highlights ambiguities (type II) that can obscure unique IR destinations, suggesting avenues for future exploration in excited states, S-matrix formulations, and ADE-related extensions.

Abstract

We present evidence for the existence of infinitely-many new families of renormalisation group flows between the nonunitary minimal models of conformal field theory. These are associated with perturbations by the $φ_{21}$ and $φ_{15}$ operators, and generalise a family of flows discovered by Martins. In all of the new flows, the finite-volume effective central charge is a non-monotonic function of the system size. The evolution of this effective central charge is studied by means of a nonlinear integral equation, a massless variant of an equation recently found to describe certain massive perturbations of these same models. We also observe that a similar non-monotonicity arises in the more familiar $φ_{13}$ perturbations, when the flows induced are between nonunitary minimal models.

Figures (8)

• Figure 1: The grid of minimal models ${\cal M}_{p,q}$ and a selection of the predicted flows, showing the unique sequences with $|I|=1$ and $|I|=2$, and one of the two sequences with $|I|=3$. Also shown are the lines $q{=}p$, $q{=}2p$ and $q{=}4p$.
• Figure 2: TBA (boxes) and NLIE data (solid lines) for the 'diagonal' sequence with $|I|= 1$.
• Figure 3: The sequence of nonunitary flows with $|I|=2$.
• Figure 4: One of the two sequences with $|I|= 3$.
• Figure 5: The $A_n$, $T_n$ and $D_n$ graphs, indicating in each case the node to which the driving term $r \cosh \theta$ in the massive TBA system should be attached.