Quantum order by disorder in Frustrated Spin Nanotubes

We investigate quantum order by disorder in a frustrated spin nanotube formed by wrapping a $J_1$-$J_2$ Heisenberg model at 45${}^{\circ }$ around a cylinder. Using Schwinger boson theory and Density Matrix Renormalization Group (DMRG), we have computed the ground-state phase diagram to reveal a ${\mathbb{Z}}_2$ phase in which collinear spin stripes form a right or left handed helix around the nanotube. We have derived an analytic estimate for the critical ${\eta }_c=J_1\mathrm{/}2J_2$ of the ${\mathbb{Z}}_2$ helical phase transition, which is in agreement with the DMRG results. By evaluating the entanglement spectrum and nonlocal string order parameters we discuss the topology of the ${\mathbb{Z}}_2$-helical phase.