  We investigate quantum phase transitions of the S=1/2 three-leg
antiferromagnetic spin tube with asymmetric inter-chain (rung) exchange
interactions. On the basis of the electron tube system, we propose a useful
effective theory to give the global phase diagram of the asymmetric spin tube.
In addition, using other effective theories we raise the reliability of the
phase diagram. The density-matrix renormalization-group and the numerical
diagonalization analyses show that the finite spin gap appears in a narrow
region around the rung-symmetric line, in contrast to a recent paper by
Nishimoto and Arikawa [Phys. Rev. B78, 054421 (2008)]. The numerical
calculations indicate that this global phase diagram obtained by use of the
effective theories is qualitatively correct. In the gapless phase on the phase
diagram, the numerical data are fitted by a finite-size scaling in the c=1
conformal field theory. We argue that all the phase transitions between the
gapful and gapless phases belong to the Berezinskii-Kosterlitz-Thouless
universality class.
