Spin Hamiltonian, order out of a Coulomb phase, and pseudocriticality in the frustrated pyrochlore Heisenberg antiferromagnet ${\mathrm{FeF}}_3$

${\mathrm{FeF}}_3$, with its half-filled ${\mathrm{Fe}}^{3+}3d$ orbital, hence zero orbital angular momentum and $S=5/2$, is often put forward as a prototypical highly frustrated classical Heisenberg pyrochlore antiferromagnet. By employing ab initio density functional theory, we obtain an effective spin Hamiltonian for this material. This Hamiltonian contains nearest-neighbor antiferromagnetic Heisenberg, biquadratic, and Dzyaloshinskii-Moriya interactions as dominant terms and we use Monte Carlo simulations to investigate the nonzero temperature properties of this minimal model. We find that upon decreasing temperature, the system passes through a Coulomb phase, composed of short-range correlated coplanar states, before transforming into an “all-in/all-out” (AIAO) state via a very weakly first-order transition at a critical temperature $T_c\approx 22$ K, in good agreement with the experimental value for a reasonable set of Coulomb interaction $U$ and Hund's coupling $J_{\mathrm{H}}$ describing the material. Despite the transition being first order, the AIAO order parameter evolves below $T_c$ with a power-law behavior characterized by a pseudo “critical exponent” $\beta \approx 0.18$ in accord with experiment. We comment on the origin of this unusual $\beta$ value.

Figure 1

The structure of ${\mathrm{FeF}}_3$. Red (dark gray) spheres denote the ${\mathrm{Fe}}^{3+}$ ions with their spin indicated by a green arrow. The ${\mathrm{F}}^{-}$ ions (not shown) are located at the (shown) bends where bonds merge. (a) The AIAO state. (b) A coplanar spin configuration (for clarity, a long-range coplanar state is shown).

Figure 2

Main panel: Variation of the AIAO order parameter $\left(m\right)$ versus temperature (in units of $J_1)$, for lattices of linear size $L=4,6,8,10$. Top inset: Fourth-order Binder cumulant of $m$ versus temperature, $T$ (in units of $J_1)$, for the same lattice sizes. Left inset: Finite-size scaling of $m(t,L)$ with $\beta =0.18\left(2\right)$ and $\nu =0.60\left(2\right)$.

Figure 3

Top row: Probability distribution functions, $P\left(m_n\right),P\left(R\right)$, and $P\left(\tilde{R}\right)$, as a function of temperature $T$ and for a lattice of linear size $L=10$. The inset of panel (b) shows the $T$ dependence of $\langle R\rangle$, which displays a sharp drop at $T_c\approx 0.06$, a further indication for the discontinuous nature of the transition. Bottom row: Temperature evolution of the neutron structure factor, $S\left(\mathit{q}\right)$, in the $\left(hhl\right)$ plane as $T$ approaches $T_c$ from the paramagnetic phase. The arrows indicate the location of pinch points for the $T=0.1$ and $T=0.08$ panels (see text).