Complex magnetic interactions and critical behavior analysis in quaternary CoFeV$_{0.8}$Mn$_{0.2}$Si Heusler alloy

We investigate the magnetic behavior and critical exponents of quaternary CoFeV$_{0.8}$Mn$_{0.2}$Si Heusler alloy to understand the interactions across the Curie temperature ($T_{\rm C}$). The Rietveld refinement of the x-ray diffraction pattern with the space group F$\bar{4}$3m confirms a single-phase cubic Y-type crystal structure. The magnetic susceptibility $\chi (T)$ data show a ferromagnetic nature with a second-order phase transition from paramagnetic to ferromagnetic at $446\pm1$~K. The saturation magnetization at 5~K is determined to be 2.2~$\mu_B$/f.u., which found to be close to the Slater--Pauling rule and indicates its half-metallic nature. The values of asymptotic critical exponents ($\beta$, $\gamma$, and $\delta$) and the $T_{\rm C}$ are extracted through detailed analytical analysis including the Modified Arrott plot, the Kouvel-Fisher (K--F) method, and the Widom scaling relation. Interestingly, the estimated values of $\beta$ = 0.369 and $\gamma$ = 1.445 closely approximate the theoretical values of the 3D Heisenberg model and second-order thermodynamic phase transition across the $T_{\rm C}$. The obtained exponents lead to the collapse of renormalized isotherms, represented by the relationship between the magnetization (m) and the applied magnetic field (h), into two distinct branches above and below the $T_{\rm C}$, which validates the reliability of the analysis. Furthermore, these exponents suggest that the spin interaction follows a decay pattern of $J(r) \sim r^{-4.99}$, indicating a short-range magnetic ordering, akin to the itinerant-electron 3D Heisenberg model.