Relaxation of frustration and gap enhancement by the lattice distortion in the Delta chain

We clarify an instability of the ground state of the $\Delta$ chain against the lattice distortion that increases a strength $(\lambda)$ of a bond in each triangle. It relaxes the frustration and causes a remarkable gap enhancement; only a $6\%$ increase of $\lambda$ causes the gap doubled from the fully-frustrated case $(\lambda=1)$. The lowest excitation is revealed to be a kink-antikink bound state whose correlation length decreases drastically with $\lambda$ increase. The enhancement follows a power law, $\Delta E_{\rm gap}\sim (\lambda-1) + 1.44 (\lambda -1)^{\frac{2}{3}}$, which can be obtained from the exact result of the continuous model. This model describes a spin gap behavior of the delafossite YCuO$_{2.5}$.