Truncated Conformal Space at c=1, Nonlinear Integral Equation and Quantization Rules for Multi-Soliton States

TL;DR

This work develops a Truncated Conformal Space approach for perturbations of a $c=1$ compact boson and uses it to numerically test the nonlinear integral equation (NLIE) for sine-Gordon/massive Thirring models derived from light-cone lattice regularization. A central finding is that pure hole states must be quantized with half-integer quantum numbers to be consistent with locality at the UV fixed point, and this choice yields an impressive agreement between TCS and NLIE in the intermediate regime. In the attractive regime, the TCS data match NLIE with deviations of order $10^{-4}$–$10^{-3}$, while in the repulsive regime agreement holds for relative energies after appropriate renormalization, confirming NLIE’s interpolation between UV and IR behavior. Overall, the study establishes the viability of TCS for $c=1$ theories, clarifies the UV operator content and state counting, and strengthens the link between the $c=1$ CFT, sine-Gordon, and massive Thirring descriptions in finite-volume spectroscopy.

Abstract

We develop Truncated Conformal Space (TCS) technique for perturbations of c=1 Conformal Field Theories. We use it to give the first numerical evidence of the validity of the non-linear integral equation (NLIE) derived from light-cone lattice regularization at intermediate scales. A controversy on the quantization of Bethe states is solved by this numerical comparison and by using the locality principle at the ultra- violet fixed point. It turns out that the correct quantization for pure hole states is the one with half-integer quantum numbers originally proposed by Mariottini et al. Once the correct rule is imposed, the agreement between TCS and NLIE for pure hole states turns out to be impressive.

Figures (5)

• Figure 1: The first few energy levels in the vacuum ($m=0$) sector at $p=\frac{2}{7}$ (plotted with diamonds) for $E_{cut}=17.0$ (the dimension of the space is $4141$) and the NLIE prediction for the vacuum scaling function (shown with a continuous line).
• Figure 2: The first few energy levels in the $m=2$ sector at $p=\frac{2}{7}$ (plotted with diamonds) for $E_{cut}=20.0$ (the dimension of the space is $3917$) and the NLIE prediction for the two hole scaling functions with quantum numbers $\left( -\frac{1}{2},\frac{1}{2}\right)$, $\left( -\frac{3}{2},\frac{3}{2}\right)$ and $\left( -\frac{5}{2},\frac{5}{2}\right)$ (shown with continuous lines).
• Figure 3: The first few energy levels in the $m=4$ sector at $p=\frac{2}{7}$ (plotted with diamonds) for $E_{cut}=26.0$ (the dimension of the space is $3403$) and the NLIE prediction for the four hole scaling functions with quantum numbers $\left( -\frac{3}{2},-\frac{1}{2},\frac{1}{2},\frac{3}{2}\right)$, $\left( -\frac{5}{2},-\frac{1}{2},\frac{1}{2},\frac{5}{2}\right)$ and $\left( -\frac{5}{2},-\frac{3}{2},\frac{3}{2},\frac{5}{2}\right)$ (shown with continuous lines).
• Figure 4: The first few energy levels relative to the vacuum in the $m=2$ sector at $p=1.5$ (plotted with diamonds) for $E_{cut}=20.0$ (the dimension of the space is $4445$) and the NLIE prediction for the relative two hole scaling functions with quantum numbers $\left( -\frac{1}{2},\frac{1}{2}\right)$ and $\left( -\frac{3}{2},\frac{3}{2}\right)$ (shown with continuous lines).
• Figure 5: The first few energy levels relative to the vacuum in the $m=4$ sector at $p=1.5$ (plotted with diamonds) for $E_{cut}=22.5$ (the dimension of the space is $4149$) and the NLIE prediction for the relative two hole scaling functions with quantum numbers $\left( -\frac{3}{2},-\frac{1}{2},\frac{1}{2},\frac{3}{2}\right)$, $\left( -\frac{5}{2},-\frac{1}{2},\frac{1}{2},\frac{5}{2}\right)$ and $\left( -\frac{5}{2},-\frac{3}{2},\frac{3}{2},\frac{5}{2}\right)$ (shown with continuous lines).