A Renormalisation group for TCSA

TL;DR

The paper analyzes errors from level truncation in the Truncated Conformal Space Approach (TCSA) for boundary renormalisation group flows and shows that TCSA spectra can mimic RG flow sequences between conformal boundary conditions. For perturbations by $\phi_{(13)}$, it derives a renormalisation group equation for the TCSA coupling, predicting finite-point artefacts that evolve with truncation level, and validates these predictions against TBA data. By applying RG-improved couplings, the authors demonstrate improved cross-level consistency and better agreement with TBA spectra, highlighting important finite-$N$ corrections in TCSA. The work illuminates how truncation effects shape observed flows, discusses the need for rescaling of the Hamiltonian or strip width, and sets the stage for refining quantitative TCSA comparisons with exact methods in boundary CFT contexts.

Abstract

We discuss the errors introduced by level truncation in the study of boundary renormalisation group flows by the Truncated Conformal Space Approach. We show that the TCSA results can have the qualitative form of a sequence of RG flows between different conformal boundary conditions. In the case of a perturbation by the field phi(13), we propose a renormalisation group equation for the coupling constant which predicts a fixed point at a finite value of the TCSA coupling constant and we compare the predictions with data obtained using TBA equations.

Figures (7)

• Figure 1: The space of boundary flows in the tricritical Ising model
• Figure 2: The normalised energy gaps for the pertubation of the strip with boundary conditions $(11)$ and $(13)$ by the field $\phi_{(13)}$ at two different truncation levels. The multiplicities of the $(31)$ and $(21)$ representations are shown on the right and left for comparison
• Figure 3: The normalised gaps for the perturbation of the strip with boundary conditions $(11)$ and $(13)$ by the field $\phi_{(13)}$ with positive and negative coupling on a logarithmic scale. The positions of the approximate fixed points $(31)$, $(21)$, $(12)$ and $(11)$ are indicated by vertical lines
• Figure 4: The normalised gaps for the perturbation of the strip with boundary conditions $(11)$ and $(13)$ in the model $M_{6,7}$ by the field $\phi_{(13)}$ with negative and positive couplings each on a logarithmic scale at truncation level 16. The positions of the approximate fixed points are indicated by vertical lines.
• Figure 5: Low-lying normalised energy gaps of the Hamiltonian in the tricritical Ising model on a strip with boundary conditions $(11)$ and $(12)$ with the latter perturbed by $\phi_{(13)}$ plotted against the logarithm of the coupling constant. On the left the TCSA data is uncorrected whereas on the right it is corrected using the RG equation (\ref{['rge']}). See text for details.