  We develop a real space quantum renormalization group (QRG) to explore a
frustrated anisotropic four-leg spin-1/2 nanotube in the thermodynamic limit.
We obtain the phase diagram, fixed points, critical points, the scaling of
coupling constants and magnetization curves. Our investigation points out that
in the case of strong leg coupling the diagonal frustrating interaction is
marginal under QRG transformations and does not affect the universality class
of the model. Remarkably, the renormalization equations express that the spin
nanotube prepared in the strong leg coupling case goes to the strong plaquette
coupling limit (weakly interacting plaquettes). Subsequently, in the limit of
weakly interacting plaquettes, the model is mapped onto a 1D spin-1/2 XXZ chain
in a longitudinal magnetic field under QRG transformation. Furthermore, the
effective Hamiltonian of the spin nanotube inspires both first and second order
phase transitions accompanied by the fractional magnetization plateaus. Our
results show that the anisotropy changes the magnetization curve and the phase
transition points, significantly. Finally, we report the numerical exact
diagonalization results to compare the ground state phase diagram with our
analytical visions.
