Coexistence of vector chiral order and Tomonaga-Luttinger liquid in the frustrated three-leg spin tube in a magnetic field

The frustrated three-leg antiferromagnetic spin-$\frac{1}{2}$ tube with a weak interchain coupling in a magnetic field is investigated by means of Abelian bosonization techniques. It is clearly shown that a vector chiral long-range order and a one-component Tomonaga-Luttinger liquid coexist in a wide magnetic-field region from a state with a small magnetization to a nearly saturated one. The chiral order is predicted to still survive in the intermediate plateau state. We further predict that (even) when the strength of one bond in the three rung couplings is decreased (increased), an Ising-type quantum phase transition takes place and the chirality vanishes (no singular phenomena occur and the chiral order is maintained). Even without magnetic fields, the chiral order would also be present if the spin tube possesses easy-plane anisotropy.

Figure 1

Relationship between $\left\{{\theta }_n\right\}$ and $\left\{{\Theta }_q\right\}$. $C$ is a constant.

Figure 2

(Color) (a) Potential $V[{\Theta }_1,{\Theta }_2]$. (b) Physically relevant zone (enclosed by the green line) in the projected ${\Theta }_1-{\Theta }_2$ plane. ${\vec{K}}_n$ play the role of the primitive translation vectors. The rung-parity transformation causes ${\Theta }_1\to -{\Theta }_1$ and the rung translation by one site does ${\vec{K}}_n\to {\vec{K}}_{n+1}$.

Figure 3

(Color) Deformed potential $V_{\delta }[{\Theta }_1,{\Theta }_2]$.

Figure 4

Phase diagram of the spin-$\frac{1}{2}$ tube. $J_{\perp }$ $(<J_{\perp }^c)$ is fixed in the weak-rung-couping regime. The bosonization approach is less valid near both the lower and upper critical fields. The upper critical field here is evaluated by the energy dispersion of one magnon.