Excited states in some simple perturbed conformal field theories

TL;DR

This work develops and tests an analytic-continuation approach to generalized TBA equations for excited states in perturbed conformal field theories ${\cal M}_{2,2N+3}$, with a detailed study of the $N=2$ case ($h=5$). By mapping the Riemann surface of the scaling function via TCSA and employing complex-plane TBA, the authors identify type I and type II singular lines, introduce a desingularisation technique, and derive explicit generalized TBA equations for one- and multi-particle excited states, including $F$- and $\mu$-term finite-volume corrections. The results are shown to reproduce infrared and ultraviolet asymptotics predicted by field theory and to align with truncated conformal space data, offering a framework likely applicable to a broad class of diagonal scattering theories. The study reveals rich structure in the crossover from ultraviolet to infrared, including transitions in the number and location of active singularities and the emergence of higher-sheet excitations.

Abstract

The method of analytic continuation is used to find exact integral equations for a selection of finite-volume energy levels for the non-unitary minimal models $M_{2,2N+3}$ perturbed by their $\varphi_{13}$ operators. The N=2 case is studied in particular detail. Along the way, we find a number of general results which should be relevant to the study of excited states in other models.

Figures (13)

• Figure 1: The first five sheets of the Riemann surface of the scaling function $F(r)$, showing on the left the connectivity of the various sheets, and on the right the positions of the square-root branch points over the complex $r$-plane.
• Figure 2: TCSA data for real $r$ between $0$ and $12$, with the lines labelled both by the conformal fields found in the ultraviolet, and by the mass gaps exhibited as $r\rightarrow\infty$.
• Figure 3: TCSA data for complex $r$ on the positive-$\lambda$ line $r=\rho e^{7\pi\imath/20}$, $0\le\rho\le 12$. Three branch points can be seen; those marked $A$ and $C$ also appear on figure 1.
• Figure 4: Contour plots of $|z_a(\theta)|/(1{+}|z_a(\theta)|)$ in the complex $\theta$ plane, for the basic TBA equation with $r{=}1$. Concentric patterns of contours are signs of singularities in $L_a(\theta)$. Those corresponding to zeroes of $Y_a(\theta)$ are marked by dots $\bullet\,$; the others are zeroes of $z_a(\theta)$. The scales on the axes are in units of $\pi/5$.
• Figure 5: Contour plots of $|z_a(\theta)|/(1{+}|z_a(\theta)|)$ in the complex $\theta$ plane, for the ground-state solution to the basic TBA equation with $r{=}e^{7\pi\imath/20}$, $\rho{=}1$. Labelling as in figure 4.