Crossover between tricritical and Lifshitz points in pyrochlore FeF$_{3}$

Pyrochlore FeF$_{3}$ (pyr-FeF$_{3}$) is a Heisenberg anti-ferromagnetic (AF) with a magnetic susceptibility deviating from the Curie-Weiss law, even at the room temperature. This compound shows a transition to a long-range ordered state with all-in all-out (AIAO) spin configuration. The critical properties of this transition have remained a matter of dispute. In this work, to gain more insight into the critical properties of pyr-FeF$_{3}$, using ab initio density functional theory (DFT), we obtain spin Hamiltonian of this material under the relative volume change with respect to the experimental volume ($\frac{\Delta V}{V_0}$) from $-0.2$ to $0.2$. We show that the relevant terms in the spin Hamiltonians are the AF exchange up to third neighbors, the nearest neighbor bi-quadratic and the direct Dyzaloshinski-Moriya (DM) interactions and find how these coupling constants vary under the volume change. Then we study the effect of volume change on the finite temperature critical behavior, using classical Monte Carlo (MC) simulation. We show that the spin system undergoes a weakly first order transition to AIAO at small volumes which turns to a second order transition close to the experimental structure. However, increasing $\frac{\Delta V}{V_0}$ to $\sim0.2$, systems shows a transition to a modular spin structure. This finding suggests the existence of a Lifshitz point in pyr-FeF$_{3}$ and may explain the unusual critical exponents observed for this compound.