Elastic constants and ultrasound attenuation in the spin-liquid phase of Cs$_2$CuCl$_4$

The spin excitations in the spin-liquid phase of the anisotropic triangular lattice quantum antiferromagnet Cs$_2$CuCl$_4$ have been shown to propagate dominantly along the crystallographic $b$ axis. To test this dimensional reduction scenario, we have performed ultrasound experiments in the spin-liquid phase of Cs$_2$CuCl$_4$ probing the elastic constant $c_{22}$ and the sound attenuation along the $b$ axis as a function of an external magnetic field along the $a$ axis. We show that our data can be quantitatively explained within the framework of a nearest-neighbor spin-$1/2$ Heisenberg chain, where fermions are introduced via the Jordan-Wigner transformation and the spin-phonon interaction arises from the usual exchange-striction mechanism.