Order Parameters and Phase Diagram of Multiferroic RMn2O5

\def\rhov{{\mbox{\boldmath{$\rho$}}}} \def\tauv{{\mbox{\boldmath{$\tau$}}}} \def\Lambdav{{\mbox{\boldmath{$\Lambda$}}}} \def\sigmav{{\mbox{\boldmath{$\sigma$}}}} \def\xiv{{\mbox{\boldmath{$\xi$}}}} \def\chiv{{\mbox{\boldmath{$\chi$}}}} \def\oh{{\scriptsize 1 \over \scriptsize 2}} \def\ot{{\scriptsize 1 \over \scriptsize 3}} \def\of{{\scriptsize 1 \over \scriptsize 4}} \def\tf{{\scriptsize 3 \over \scriptsize 4}} Recently there has been great interest in systems which display phase transitions at which incommensurate magnetic order and a spontaneous polarization develop simultaneously. Perhaps the most puzzling and seemingly complicated behavior occurs in the series of compounds RMn$_2$O$_5$, where R=Y, Ho, Er, Tb, Tm, and Dy. (For references to experimental data, see [1].) The sequence of magnetoelectric phases of the type I systems R=Tb, Ho, and Dy is slightly different from that of the type II systems R= Y, Tm, and Er. At about 45K both types develop essentially collinear modulated magnetic order into a ``high-temperature ordered" (HTO) phase with a wave vector ${\bf q} = (1/2-\delta , 0, 1/4 + \epsilon)$ where $\delta$ and $|\epsilon|$ are of order 0.01 and the spontaneous polarization is zero. There is a lower-temperature phase transition to a ferroelectric phase in which transverse magnetic order appears and produces a magnetic spiral with $\delta=\epsilon=0$. In type I systems, this transition occurs directly from the HTO phase, whereas for type II systems, there is an intervening ferroelectric phase in which $\epsilon=0$, but $\delta$ remains nonzero. %At low ($<10$K) temperature the classification into types I and II %breaks down and each system requires its own specific description. I will discuss a Landau free energy[1] which allows both type I and type II sequences of phase transitions. This theory is couched in terms of the uniform polarization vector ${\bf P}$ and two complex-valued magnetic order parameters $\sigma_1({\bf q})$ and $\sigma_2 ({\bf q})$ whose symmetry follows from the magnetic structure analyses.[2] The magnetoelectric coupling and the competition between commensurate and incommensurate phases are analyzed. \\[4pt] [1] A. B. Harris, A. Aharony, and O. Entin-Wohlman, Phys. Rev. Lett. {\bf 100}, 217202 (2008) and J. Phys. Condens. Mat. {\bf 20}, 434202 (2008). \\[0pt] [2] A. B. Harris, Phys. Rev. {\bf 76}, 054447 (2007); A. B. Harris, M. Kenzelmann, A. Aharony, and O. Entin-Wohlman, Phys. Rev. B {\bf 78}, 014407 (2008).