Conformal field theory in the Tomonaga-Luttinger model with the 1/rβ long-range interaction

We attempt to construct U(1) conformal field theory (CFT) in the Tomonaga-Luttinger (TL) liquid with the 1/r(beta) long-range interaction (LRI). Treating the long-range forward scattering as a perturbation and applying CFT to it, we derive finite size scalings which depend on the power of the LRI. The obtained finite size scalings give nontrivial behaviours when beta is odd and is close to 2. We find consistency between the analytical arguments and numerical results in the finite size scaling of energy.