Spin-chirality separation and $S_3$ symmetry breaking in the magnetization plateau of the quantum spin tube

We study the magnetization plateau state of the three-leg spin-$\frac{1}{2}$ tube in the strong rung coupling region, where $S_3$ symmetry breaking and the low-energy chirality degree of freedom play crucial roles. On the basis of the effective chirality model and density matrix renormalization group, we clarify that, as the leg coupling increases, the chirality liquid with gapless nonmagnetic excitations, the spin-imbalance phase, and the vector-spin-chirality ordered phase emerge without closing the plateau spin gap. The relevance of these results to experiments is also discussed.

Figure 1

(a) Structure of the three-leg spin tube and (b) ground-state phase diagram of the $\frac{1}{3}$ plateau state of the spin tube. The plateau is predicted to vanish at a strong leg-coupling point $J/J^{\prime }\sim 0.1$[11, 12] (see the text).

Figure 2

Longitudinal spin correlation function $\langle S_{i,j}^z{S}_{i,j^{\prime }}^z\rangle$ of the spin tube for $J^{\prime }=0.01$, 0.1, 0.2, 0.3, 0.4, and 0.45 from bottom to top, where $m=\frac{1}{6}$. Two solid lines indicate guides for $\eta =0.5$ (XY chain) and 0.1.

Figure 3

Expectation value of the order parameters $\chi$ (solid circle) and $\mu$ (open circle). (Inset) The order parameters around the transition points. The vertical broken lines indicate the transition points $J_{c1}^{\prime }$ and $J_{c2}^{\prime }$.

Figure 4

Spin profile $\langle S_{i,j}^z\rangle$ of the spin-imbalance phase. The tube length is $L=96$ and the intertriangle coupling is $J^{\prime }=0.485$. Solid circles denote the expectation value of the most polarized spin in each rung triangle and the open circles correspond to those of remaining two spins on the triangle. The horizontal broken lines is the averaged magnetization of each rung in the plateau state.

Figure 5

Correlation function $|\langle S_{i,j}^z{S}_{i,j^{\prime }}^z\rangle -m_i^2|$ in the spin-imbalance phase. The solid circles is the correlator for the chain consisting of the most polarized spins on the unit triangules, and the open circles correspond to that for the remaining two chains. (Inset) The $J^{\prime }$ dependence of the inverse correlation length ${\xi }^{-1}$ along the chains for the most and less polarized spins in the triangle.

Figure 6

Expectation value of the projection operator $\langle {\widehat{P}}^{1/2}\rangle$ on the unit triangle.