The nature of the $\beta$-peak in the loss modulus of amorphous solids

Glass formers exhibit, upon an oscillatory excitation, a response function whose imaginary and real parts are known as the loss and storage moduli respectively. The loss modulus typically peaks at a frequency known as the $\alpha$ frequency which is associated with the main relaxation mechanism of the super-cooled liquid. In addition, the loss modulus is decorated by a smaller peak, shoulder or wing which is referred to as the $\beta$-peak. The physical origin of this secondary peak had been debated for decades, with proposed mechanisms ranging from highly localized relaxations to entirely cooperative ones. Using numerical simulations, we expose a clear and unique cooperative mechanism for the said $\beta$-peak which is distinct from that of the $\alpha$-peak.