Ultrasound evidence for multicomponent superconducting order parameter in Ba$_{1-x}$K$_x$Fe$_2$As$_2$ with electron quadrupling phase

Experiments have pointed to the formation of the electron quadrupling condensate in Ba$_{1-x}$K$_x$Fe$_2$As$_2$ at $x \sim 0.8$. The state spontaneously breaks time-reversal symmetry and is sandwiched between two critical points, separating it from the broken time-reversal symmetry (BTRS) superconducting state at $T_{\rm c}^{U(1)}$ and normal-metal state at $T_{\rm c}^{\rm Z2}$. We report a theory of the acoustic effects spectroscopy of systems with an electron quadrupling phase based on ultrasound-velocity measurements. We show that the experimental results are consistent with BTRS superconductivity at $x \sim 0.8$, fulfilling the necessary condition for the formation of electron quadrupling in Ba$_{1-x}$K$_x$Fe$_2$As$_2$. We provide the theoretical basis and the experimental strategy to study the order parameter symmetry of emerging quadrupling condensates in superconductors.