Excited state TBA and renormalized TCSA in the scaling Potts model

TL;DR

This work develops and validates a dual framework for the finite-volume spectrum of the scaling Potts model by coupling renormalized TCSA with an excited-state TBA system, covering both paramagnetic and ferromagnetic phases. It establishes a systematic renormalization approach, constructs counter terms for ground and excited states, and derives UV and IR checks that align with conformal field theory and exact S-matrix predictions. The authors demonstrate, through detailed numerical comparisons, that the renormalized TCSA reproduces TBA results with high precision across a wide range of volumes and provides accurate two-particle phase shifts. The study has broader implications for perturbed CFTs and potential extensions to theories with irrelevant operators and to related areas such as AdS/CFT and non-equilibrium physics.

Abstract

We consider the field theory describing the scaling limit of the Potts quantum spin chain using a combination of two approaches. The first is the renormalized truncated conformal space approach (TCSA), while the second one is a new thermodynamic Bethe Ansatz (TBA) system for the excited state spectrum in finite volume. For the TCSA we investigate and clarify several aspects of the renormalization procedure and counter term construction. The TBA system is first verified by comparing its ultraviolet limit to conformal field theory and the infrared limit to exact S-matrix predictions. We then show that the TBA and the renormalized TCSA match each other to a very high precision for a large range of the volume parameter, providing both a further verification of the TBA system and a demonstration of the efficiency of the TCSA renormalization procedure. We also discuss the lessons learned from our results concerning recent developments regarding the low-energy scattering of quasi-particles in the quantum Potts spin chain.

Figures (5)

• Figure 5.1: (color online) Comparing TCSA and TBA for the ground state. The slow convergence of the TCSA is apparent from the raw data; renormalized data are only presented for level $12$, as the others would not be discernible on the plot. This plot does not have the bulk energy subtracted to show that the renormalization also gives back the right value for the universal bulk energy term.
• Figure 5.2: (color online) Comparing TCSA and TBA for excited states. PM stands for paramagnetic, FM for ferromagnetic phase, GS means ground state (twisted in the ferromagnetic phase). The paramagnetic $AA$ and twisted ferromagnetic $A\bar{A}$ are so close numerically that they eventually overlap at this resolution.
• Figure 5.3: (color online) Comparing $\delta_{A\bar{A}}(\theta)$ extracted from the third excited TCSA level in sector $\mathcal{H}_{0}$ (lowest lying moving $A\bar{A}$ state) to the scattering theory predictions (\ref{['eq:Potts_BY_paramagnetic']},\ref{['eq:Potts_BY_ferromagnetic']}) and to TBA.
• Figure 5.4: (color online) Comparing $\delta_{A\bar{A}}(\theta)$ extracted from the first excited TCSA level in sectors $\mathcal{H}_{\pm}$ in the ferromagnetic phase (lowest lying twisted $A\bar{A}$ state) to the scattering theory predictions (\ref{['eq:Potts_BY_ferromagnetic']}) and to TBA.
• Figure 5.5: (color online) Comparing $\delta_{AA}(\theta)$ extracted from the first excited TCSA level in sectors $\mathcal{H}_{\pm}$ in the paramagnetic phase (lowest lying $AA$/$\bar{A}\bar{A}$ state) to the scattering theory predictions (\ref{['eq:Potts_BY_paramagnetic']}) and to TBA.