Magnetoelastic couplings in the deformed Kagom\'{e} quantum spin lattice of volborthite

Microscopic spin interactions on a deformed Kagom\'{e} lattice of volborthite are investigated through magnetoelastic couplings. A negative longitudinal magnetostriction $\Delta L<0$ in the $b$ axis is observed, which depends on the magnetization $M$ with a peculiar relation of $\Delta L/L \propto M^{1.3}$. Based on the exchange striction model, it is argued that the negative magnetostriction originates from a pantograph-like lattice change of the Cu-O-Cu chain in the $b$ axis, and that the peculiar dependence arises from the local spin correlation. This idea is supported by DFT+$U$ calculations simulating the lattice change and a finite-size calculation of the spin correlation, indicating that the recently proposed coupled-trimer model is a plausible one.