Hysteretic phenomena in multiferroic HoMnO$_{3}$

The hexagonal HoMnO$_{3}$ is multiferroic with ferroelectric transition at T$_{C}$=875 K, the Mn$^{3+}$ moments order at T$_{N}$=72 K, Mn$^{3+}$- spin reorientation transitions occur at T$_{sp1}$=34 K and T$_{sp2}$=4.9 K, and magnetic Ho$^{3+}$ orders at T$_{Ho }\sim $ 5.2 K. Our dc susceptibility measurements of HoMnO$_{3}$ single crystal under both zero-field cooled (ZFC) and field-cooled (FC) conditions at low applied magnetic fields reveal un unusual anomaly. In particular, below 4.9 K, the out-of-plane susceptibility ($c$-axis magnetization) is higher in the case of a ZFC measurement compared with that of a FC measurement. Above 4.9 K, the ZFC curve is lower than the FC curve, as well as for the in-plane susceptibility measurement in the temperature range below T$_{N}$=34 K. We assume that these phenomena are related to the magnetic and ferroelectric domain boundary structures and sensitively respond to changes of the magnetic structure such as spin-rotations at the phase transitions. We also have observed a sizable difference in the magnetization of HoMnO$_{3}$ polycrystalline samples with nano and micro size particles.