Universal critical behavior in the ferromagnetic supercondutor Eu(Fe$_{0.75}$Ru$_{0.25}$)$_{2}$As$_{2}$

The study of universal critical behavior is a crucial issue in a continuous phase transition, which groups various critical phenomena into the universality classes. The understanding on the nature of magnetism in Eu-based ferromagnetic superconductors is largely impeded by the infeasibility of performing inelastic neutron scattering measurements to deduce the microscopic magnetic exchange couplings. However, by systemically combining the neutron diffraction experiment, the first-principles calculations, and the quantum Monte Carlo simulations, we have obtained a perfectly consistent universal critical exponent value of $\beta=0.385(10)$ experimentally and theoretically for Eu(Fe$_{0.75}$Ru$_{0.25}$)$_{2}$As$_{2}$, from which the magnetism in the Eu-based ferromagnetic superconductors is identified as the universal class of three dimensional anisotropic quantum Heisenberg model with long-range magnetic exchange coupling. This study not only clarifies the nature of microscopic magnetic behaviours in the Eu-based ferromagnetic superconductors, but also methodologically opens a new avenue for studying the universal critical behaviors asscociated with magnetic phase transitions.