Multiple magnetic transitions on the pentagonal Cairo lattice in Bi$_4$Fe$_5$O$_{13}$F

Pentagonal Cairo lattice gives rise to an unusual frustrated scenario that was actively studied theoretically but still lacks experimental manifestation in real materials. In this talk, I will present a novel Cairo-lattice antiferromagnet Bi$_4$Fe$_5$O$_{13}$F that features a sequence of three magnetic transitions at $T_1=62$\,K, $T_2=71$\,K, and $T_N=178$\,K. Using a combination of neutron diffraction and M\"ossbauer spectroscopy, we show that the magnetic structure below $T_1$ is consistent with orthogonal spin order anticipated by theory. The collinear magnetic structure observed between $T_1$ and $T_2$ is also anticipated by theory, but only for the quantum version of the model, whereas Bi$_4$Fe$_5$O$_{13}$F with its spin-$\frac52$ Fe$^{3+}$ should be close to the classical limit. Finally, the magnetic structure between $T_2$ and $T_N$ is again orthogonal, but individual spin directions are different from those below $T_1$. A microscopic magnetic model of Bi$_4$Fe$_5$O$_{13}$F will be presented, and possible origins of the spin-reorientation transitions in this material will be discussed.