Low-energy effective theory and two distinct critical phases in a spin-12frustrated three-leg spin tube

Abstract

Motivated by the crystal structures of [(CuCl${}_{2}$tachH)${}_{3}$Cl]Cl${}_{2}$ and Ca${}_{3}$Co${}_{2}$O${}_{6}$, we develop a low-energy effective theory using the bosonization technique for a spin-$\frac{1}{2}$ frustrated three-leg spin tube with trigonal prism units in two limit cases. The features obtained with the effective theory are numerically elucidated by the density matrix renormalization group method. Three different quantum phases in the ground state of the system, say, one gapped dimerized phase and two distinct gapless phases, are identified, where the two gapless phases are found to have the conformal central charge $c=1$ and $\frac{3}{2}$, respectively. Spin gaps, spin and dimer correlation functions, and the entanglement entropy are obtained. In particular, it is disclosed that the critical phase with $c=\frac{3}{2}$ is the consequence of spin frustrations, which might belong to the SU(2)${}_{k=2}$ Wess-Zumino-Witten-Novikov universality class, and is induced by the twist term in the bosonized Hamiltonian density.