  We discuss effects of $1/r^\beta$ type long-range (LR) interactions in a
tight-binding model by utilizing the bosonization technique, renormalization
group and conformal field theory (CFT). We obtain the low energy action known
for Kibble's model which generates the mass gap in 3 dimension when $\beta =1$,
the Coulomb force case. In one dimension, the dispersion relations predict that
the system remains gapless even for $\beta =1$ and the existences of
Tomonaga-Luttinger liquid (TLL) when $\beta > 1$. When $\beta =1$, the LR
interactions break TLL in the long wavelength limits, even if the strength is
very small. We make the more precise arguments from the stand point of the
renormalization group and CFT. Finally we derive the accurate finite size
scaling of energies and thermodynamics. Moreover we proceed to numerical
calculations, considering the LR umklapp process terms. We conclude that the
TLL phase become wider in the strength space of interactions as the power
$\beta$ approaches to 1.
