Conformal field theory in Tomonaga-Luttinger model with $1\mathrm{/}{r}^{\beta }$ type long-range interactions

The validity of the CFT in the Tomonaga-Luttinger liquid with the $1\mathrm{/}{r}^{\beta }$ type long-range interactions is discussed. The arguments by CFT for long-range forward scatterings predict the finite size corrections which depend on the power of the long-range interactions. Especially from finite size corrections in the ground state, we find the effective central charge deviates from $c=1$. These conformal arguments are numerically tested in the one particle excitations and the ground state.