Nonlinear Integral Equation and Finite Volume Spectrum of Sine-Gordon Theory

TL;DR

The paper derives a corrected nonlinear integral equation (NLIE) for the sine-Gordon/massive Thirring finite-volume spectrum from a light-cone lattice, clarifying subtleties with special roots and the second determination. It shows that ultraviolet consistency with the c=1 UV limit requires a refined quantization rule δ_eff = 0, and demonstrates that infrared scattering data emerge correctly from the NLIE via dressed Bethe Ansatz structures. By comparing NLIE predictions with Truncated Conformal Space (TCS) results, the authors validate the NLIE as a description of the finite-volume spectrum for even topological charge, while highlighting that only states consistent with locality in the UV should be taken as physical. The study thus provides a unified framework for all excited states across coupling, clarifies the role of complex roots and holes, and establishes NLIE as a robust tool for finite-size analyses in integrable 1+1D field theories.

Abstract

We examine the connection between the nonlinear integral equation (NLIE) derived from light-cone lattice and sine-Gordon quantum field theory, considered as a perturbed c=1 conformal field theory. After clarifying some delicate points of the NLIE deduction from the lattice, we compare both analytic and numerical predictions of the NLIE to previously known results in sine-Gordon theory. To provide the basis for the numerical comparison we use data from Truncated Conformal Space method. Together with results from analysis of infrared and ultraviolet asymptotics, we find evidence that it is necessary to change the rule of quantization proposed by Destri and de Vega to a new one which includes as a special case that of Fioravanti et al. This way we find strong evidence for the validity of the NLIE as a description of the finite size effects of sine-Gordon theory.

Figures (6)

• Figure 1: Contour for the integration. The crosses are roots while the circles are holes.
• Figure 2: Values of the function $f(x)$; $y$ is a special object and $\nu$ is a positive real number.
• Figure 3: The typical behaviour of the counting function $Z$ when a) $z_{1}$ is normal and when b) $z_{1}$ is special.
• Figure 4: The analytic structure of the counting function if $\sigma >\frac{\pi }{2}$. The logarithmic cuts are indicated with the wiggly lines. The one lying in the upper half plane originates from the root in the lower half plane and vice versa. We also indicated the value of the discontinuity across the cuts.
• Figure 5: The UV behaviour of $Z$ for a) $\gamma >\frac{\pi }{2}$ and b) $\gamma <\frac{\pi }{2}$. In b), the root at the origin turned into a special one and two normal holes appeared at the points where $Z$ crosses the horizontal axis with a positive derivative.