The dielectric constant of multiferroic hexagonal HoMnO 3 exhibits an unprecedented diversity of anomalies at low temperatures ͑1.8 K Ͻ T Ͻ 10 K͒ and under external magnetic fields related to magnetic phase transitions in the coupled system of Ho moments, Mn spins, and ferroelectric polarization. The derived phase diagram is far more complex than previously assumed including reentrant phases, phase transitions with distinct thermal and field hysteresis, as well as several multicritical points. Magnetoelastic interactions introduce lattice anomalies at the magnetic phase transitions. Among the multiferroic compounds hexagonal rare-earth manganites, RMnO 3 ͑R = Sc, Y, Ho to Lu͒, have attracted recent attention because of the coexistence of antiferromagnetic ͑AFM͒ and ferroelectric ͑FE͒ orders below their Néel temperatures, T N Ͻ 100 K. The mutual interference and the possible correlation of magnetic and ferroelectric orders are of fundamental interest and of significance for prospective applications. Since the report 1 of a dielectric anomaly at T N of YMnO 3 and later observations of similar effects in other hexagonal RMnO 3 the origin of the magnetodielectric coupling in these compounds has been a matter of discussion.
2-5In the magnetic symmetries realized just below T N the direct coupling between the in-plane staggered AFM magnetization and the c-axis FE polarization is not allowed. 6 The indirect coupling via lattice strain or other secondary effects has therefore been proposed. 5 In fact, the existence of strong spin-phonon interactions was concluded from the anomalous temperature dependence of the thermal conductivity 7 and the thermal expansion anomalies observed at T N of HoMnO 3 . 8 The magnetic order below 100 K is very complex for some RMnO 3 and it is determined by the strong in-plane AFM superexchange interaction of the Mn 3+ spins and their geometric frustration in the P6 3 cm hexagonal structure. While strong in-plane magnetic correlations exist well above the Néel temperature 8,9 the long-range AFM order of the Mn 3+ sets in only below 100 K with a frustrated, noncollinear spin arrangement ͑neighboring moments form an angle of 120°͒. The different spin arrangements and magnetic symmetries compatible with the P6 3 cm crystalline structure have been discussed in detail by Fiebig et al. 10 The direct magnetic exchange of the R 3+ ions, essential at low temperatures, and the coupling of the rare-earth moments with the Mn spins as well as the FE polarization gives rise to a wealth of interesting physical phenomena and a complex magnetic phase diagram. Several subsequent magnetic phase changes have been observed in HoMnO 3 below T N = 76 K, including a sharp Mn spin-rotation transition at T SR = 33 K with the onset of AFM Ho-moment order and another transition at about 5 K involving a major increase of the Ho 3+