Soliton binding and low-lying singlets in frustrated odd-legged $S=\frac{1}{2}$ spin tubes

Motivated by the intriguing properties of the vanadium spin tube ${\mathrm{Na}}_2{\mathrm{V}}_3{\mathrm{O}}_7$, we show that an effective spin-chirality model similar to that of standard Heisenberg odd-legged $S=\frac{1}{2}$ spin tubes can be derived for frustrated inter-ring couplings, but with a spin-chirality coupling constant $\alpha$ that can be arbitrarily small. Using density matrix renormalization group and analytical arguments, we show that, while spontaneous dimerization is always present, solitons become bound into low-lying singlets as $\alpha$ is reduced. Experimental implications for strongly frustrated tubes are discussed.

Figure 1

Two models for ${\mathrm{Na}}_2{\mathrm{V}}_3{\mathrm{O}}_7$. (a) Nonfrustrated, nine-leg spin tube. (b) Frustrated, three-leg spin tube.

Figure 2

Extrapolated excitation gaps in the four $[S^z,{\tau }^z]$ sectors as a function of $\alpha$. Inset: Power-law fits of the small $\alpha$ results.

Figure 3

Nearest-neighbor expectation values of spin and chirality along a chain with 50 sites in the sectors $[S^z=0,{\tau }^z=0]$, [0, 1] and [1, 0] for $\alpha =0.1$. The ground state (a) is dimerized. The unaltered dimer patterns and the constrictions at the center of the chain in (b) and (c) indicate bound soliton states, whereas the two domain walls in (d) suggest the presence of two unbound solitons.

Figure 4

Specific heat per site as a function of the temperature $T$ for $\alpha =0.2$ and 1.0 calculated by exact diagonalization of systems with up to eight sites.