  The ferroelectric phase of the multiferroic hexagonal manganites (h-ReMnO3)
has been reported to undergo a series of magnetic transitions involving
long-range ordering/reorientation of 4fRe3+ and/or 3dMn3+ spins below room
temperature. These transitions have attracted a lot of attention in recent
years due to the geometrically frustrated nature of magnetic interactions. We
have revisited these transitions in high quality single crystals of h-DyMnO3
using dc and ac susceptibility measurements as a function of temperature (T),
magnetic field (H) and frequency (w) supplemented by specific heat
measurements. Taking h-DyMnO3 as an example, we show that the Dy3+ spins below
TN~68K are in a superferrimagnetic (SFIM) state whereas they undergo spin-glass
(SG) transition below TDy3+~7K. Our observations demonstrate that neither the
N\'eel transition at TN~68K nor the transition at TDy3+~7K is associated with
long-range ordered states of Dy3+ spins as believed so far in the literature.
The SG state of h-DyMnO3 is quite exotic as it occurs in an ordered compound
purely due to geometrical frustration without any random disorder. Further, it
shows an interesting crossover from de Almeida-Thouless type exponent (m=2/3)
to Gabay-Toulouse type (m=2) with increasing field which cannot be explained in
terms of the existing mean field theories of SG transition in Ising or
Heisenberg systems but is expected for a vector X-Y SG system. Our observations
call for a systematic reinvestigation of the nature of magnetic transitions
involving Re3+ ions in other h-ReMnO3 also.
