Exchange topology and criticality in ferrite and chromium spinels: a unified Monte Carlo analysis
We report a unified analysis of Metropolis Monte Carlo results for two families of magnetic spinels: inverse ferrites Fe$^{3+}_{A}$[M$^{2+}$Fe$^{3+}$]$_{B}$O$_4$ (M = Co, Cu, Fe, Ni), where superexchange couples two chemically distinct sublattices, and chromium spinels $A$Cr$_2X_4$ ($A$ = Zn, Cd, Hg, $X$ = S, Se) together with the breathing-lattice chromates Li$M$Cr$_4$O$_8$ ($M$ = Ga, In), where a single Cr$^{3+}$ species occupies a corner-sharing tetrahedral network. Placing the exchange constants, transition temperatures, critical exponents, hysteresis, and magnetocaloric responses of these systems on a common footing, we introduce two reduced quantities not previously reported: the ratio $θ_{\mathrm{CW}}/T_C$ for the ferrites and the normalized ordering scale $t^{*}=k_BT_C/[J_1S(S+1)]$ for the chromium compounds. The ferrites cluster in the range $θ_{\mathrm{CW}}/T_C = 0.94$-$1.19$, close to the mean-field expectation of unity and the signature of dominant, unfrustrated A-B superexchange, and their exponents ($β= 0.20$-$0.26$, $γ= 1.23$-$1.27$, $δ= 4.76$-$4.78$) follow the three-dimensional Ising class. The chromium systems split into three regimes: $t^{*} \approx 1.4$-$1.9$ for Ising-treated sulfides, $t^{*} \approx 0.99$ for Heisenberg-treated selenides, and $t^{*} \approx 0.24$-$0.25$ for the antiferromagnetic breathing chromates, quantifying the combined suppression of $T_C$ by continuous spin symmetry and by geometric frustration. Finite-thickness simulations of Fe$_3$O$_4$ resolve a dimensionality crossover between two and four unit cells. We identify the Ising-versus-Heisenberg dependence of the predicted universality class in frustrated chromites as the principal open problem.