The renormalisation group for the truncated conformal space approach on the cylinder

TL;DR

This paper extends the truncated conformal space approach (TCSA) to bulk perturbations on the cylinder, deriving the leading truncation-dependent corrections as a ground-state divergence, a coupling-constant renormalisation, and a representation-dependent energy rescaling. It provides explicit perturbative expressions for these corrections and demonstrates that incorporating them dramatically improves TCSA predictions for the tri-critical Ising model, including substantial convergence of ground-state energies and excitation gaps, and corrects for large divergences when $h>1/2$. A key finding is that perturbations with conformal weight $h>3/4$ cause the spectrum to fail to converge, though ratios of energy gaps still converge, while on the strip geometry no simple universal renormalisation suffices. The results offer a practical RG framework to extend TCSA’s applicability, including to irrelevant perturbations, using only standard CFT data such as scaling dimensions, OPE coefficients, and modular S-matrix.

Abstract

In this paper we continue the study of the truncated conformal space approach to perturbed conformal field theories, this time applied to bulk perturbations and focusing on the leading truncation-dependent corrections to the spectrum. We find expressions for the leading terms in the ground state energy divergence, the coupling constant renormalisation and the energy rescaling. We apply these methods to problems treated in two seminal papers and show how these RG improvements greatly increase the predictive power of the TCSA approach. One important outcome is that the TCSA spectrum of excitations is predicted not to converge for perturbations of conformal weight greater than 3/4, but the ratios of excitation energies should converge.

Figures (8)

• Figure 1: The ground state energy for the model $\mathcal{M} A_4^{(+)}$ at truncation levels 3 (dotted), 4 (dashed), 5 (solid) and 9 (dot-dashed) plotted against $r$ together with the leading exact IR behaviour (thin solid line).
• Figure 2: The first gap in the even sector of $\mathcal{M} A_4^{(+)}$ at truncation levels 3 (dotted), 4 (dashed), 5 (solid) and 9 (dot-dashed), plotted against $\lambda$.
• Figure 3: Normalised energy gaps for $\mathcal{M} A_4^{(+)}$ plotted against $\log(\lambda)$ at truncation level 9.
• Figure 4: The gaps for the massive perturbation $\mathcal{M} A_4^{(-)}$ at truncation levels 5 for figures (a) and (b) and 9 for figure (c). In all cases the approximate Bethe-Yang two- and four-particle energies are also given as dotted and dashed lines, as calculated in KM1.
• Figure 5: The scaling gaps for the massive perturbation $\mathcal{M} A_4^{(-)}$ at truncation levels 5 together with the exact TBA results for the first gap (dotted).