Interplay between spin dynamics and crystal field in multiferroic compound HoMnO${}_3$

Multiferroics continue to attract much attention in condensed matter physics, because of the promising applications in spintronics they offer, but also because these materials raise fundamental questions. One of these concerns the understanding of the microscopic origin of the coupling between magnetic and electric degrees of freedom, and how new excitations resulting from this coupling, for instance electromagnons, can emerge. Electromagnons were first reported in 2006 as magnetic excitations excitable by the electric field component of the electromagnetic wave Pimenov2006 (). Since then, electromagnons have been observed in several compounds with non colinear magnetic order such as perovskites Pimenov2006 (); Sushkov2007 (), hexaferrites Kida2009 () or even the more simple cupric oxide Jones2014 (). Various mechanisms at the origin of these excitations were invoked and identified, among which the inverse Dzyaloshinskii-Moriya interaction Katsura2007 (), symmetric exchange-striction mechanism Mochizuki2010b () or higher harmonics of the magnetic cycloids Mochizuki2010 (). Other types of excitations involving electric and magnetic degrees of freedom have also been reported such as atomic rotation mode excited by the magnetic field component of the terahertz (THz) light Chaix2013 () or hybrid excitations arising from the interplay between rare-earth crystal field (CF) excitations and 3d transition metal magnons Sirenko2008 (); Kang2010 (), in particular in the hexagonal manganite ErMnO${}_3$ Chaix2014 (). All these hybrid excitations often lie in the THz range and are also investigated using inelastic neutron Pailhes2009 () or X ray scattering Toth2016 () and Raman spectroscopy Aupiais2018 ().

The hexagonal manganites h-RMnO${}_3$  where R= Y, Er, Ho, etc, crystallizing in the P6${}_3$cm space group, have been widely studied Bertaut1963 (); Katsufuj2002 (); Cheong2007 (); Sim2016 () since they combine remarkable properties associated to multiferroism and magnetic frustration. They belong to type I multiferroics, where ferroelectricity and magnetic orders occur at different temperatures: onset of an electric polarization along the c-axis at much higher temperature (T${}_C$$\simeq$800K) than the ordering of the Mn magnetic moments (T${}_N$$\simeq$80K). Despite these different energy scales, non negligible coupling between the electric and magnetic degrees of freedom are present. For instance, the Mn magnetic ordering is accompanied by ionic motions inside the unit cell Fabreges2009 (); Lee2008 () giving rise to a giant magneto-elastic coupling, connected with an increase of the ferroelectric polarization. The Mn sublattice consists of triangular planes (Fig. 1) with a peculiar stacking along the c-axis, each Mn${}^{3+}$ ion sitting close to the middle of the triangle of Mn${}^{3+}$ belonging to adjacent layers. The actual materials have then in common a 120${}^{\circ }$ spin structure for the Mn magnetic moments in the triangular planes, but the stacking and relative phase difference of these patterns depend on the R atom ((magnetic R=Ho, Yb, or not R= Y, Sc), and correspond to the four one-dimensional irreducible representations shown in Fig. 1. A key parameter tuning the sign of the exchange interaction and the Mn${}^{3+}$ single-ion anisotropy seems to be the shift away from 1/3 of the $x$ position of the Mn${}^{3+}$ ions within the triangular plane Fabreges2009 (); Solovyev2014 (). The ions R${}^{3+}$ are situated on two Wyckoff sites 2a and 4b. For magnetic rare earths, they also order at lower temperature, either polarized by the Mn molecular field (site 4b) or from their mutual interaction in a temperature range of few K (site 2a) Fabreges2008 (); Chattopadhyay2018 ().

In this study, we focus on the Ho compound, which has been intensively studied for its rich $H-T$ phase diagram and its strong magnetoelectric effects Cheong2007 (); Lottermoser2004 (). It becomes ferroelectric at 875 K and the Mn magnetic moments order at T${}_N$=75 K in the ${\Gamma }_4$ representation (see Fig. 1). Remarkably, a spin reorientation occurs at T${}_{SR}$=37 K which is concomitant with the ordering of the Ho(4b) magnetic moments, while the Ho(2a) moments order around $T_{Ho}$=5 K, both along the c-axis Munoz2001 (); Fiebig2002 (); Brown2006 (); Brown2008 (); Nandi2008 (); Fabreges2009 (). This Mn reorientation transition, unique in the hexagonal manganites, shows up also in the electric properties with a drop in the electric polarisation below T${}_{SR}$ which is further recovered at $T_{Ho}$ Hur2009 (). Signatures of the rare-earth Mn coupling in the spin dynamics in an applied magnetic field have also been reported Talbayev2008 (); Laurita2017 (), although no rare-earth CF-Mn magnon hybridization has yet been observed.

We report here a detailed study on the dynamical properties of this compound in the THz range using both electromagnetic waves and neutrons as complementary probes. We show that the THz dynamics is particularly complex in HoMnO${}_3$ due to the presence of several Ho CF excitations in addition to the Mn spin waves that surprisingly become non-dispersive along the c-axis at low temperature. To interpret the observed spectra and their remarkable temperature dependence through the spin reorientation transition and below, we have performed calculations both for Mn and Ho dynamics. Our work brings new insight on the complex spin dynamics of this compound including possible signatures of the couplings between the different magnetic ions, but also with their lattice surroundings.

Single crystals have been grown using the floating zone method, yielding rods of about 7 mm in diameter and 4 cm long. From these crystals, two small plaquettes were cut for the THz measurements (Typically 2 mm in diameter and 300 $\mu$m in thickness): one with the c-axis perpendicular to the plaquette, the other one with the c-axis within the plaquette. The polarized THz emission of the synchrotron at SOLEIL was used to probe the THz properties similarly to the study on ErMnO${}_3$ Chaix2014 ().

Inelastic neutron scattering (INS) measurements were performed at the thermal spectrometer IN8 (ILL) as well as at the 4F1 and 4F2 cold triple-axes spectrometers installed at LLB. The final wave vector used was $k_f$ = 2.662, 1.55 or 1.3 Å${}^{-1}$ depending on the desired energy resolution (0.8, 0.25 and 0.12 meV respectively). Higher order reflections were removed using a PG (IN8) or nitrogen cooled Beryllium filter (4F). The sample was mounted to have access to $Q=(h,0,\ell )$ scattering wave-vectors and attached to the cold finger of an orange cryostat. Several Energy-scans taken along $Q=(h,0,\ell )$ were collected to construct the maps shown in Fig. 2 and 3.

The magnetic excitations have been first investigated by INS. Figures 2a and 3 give an overview of those results, showing the spectra along $a^{\ast }$, along $c^{\ast }$, and the temperature dependence measured at the zone centers (1, 0, 0) and (1, 0, 1). Dispersive spin waves are clearly visible along $a^{\ast }$ with a maximum of the branches around 17 meV (140 cm${}^{-1}$) while the amplitude of dispersion along $c^{\ast }$ is much weaker, reflecting the hierarchy of the magnetic interactions much stronger within the ab-planes than in between. In addition to the spin waves, non-dispersive excitations are visible corresponding to transition between CF levels of the two Ho${}^{3+}$ ions on 4b and 2a sites. Remarkably, a significant temperature dependence of the magnetic excitations is visible at T${}_{SR}$ as shown at the zone centers (1, 0, 0) and (1, 0, 1) in Fig.3 where changes of spectral weight are observed for all the excitations below 30 cm${}^{-1}$. More surprisingly, along $c^{\ast }$, the spin waves dispersion disappears gradually below T${}_{SR}$ and finally the corresponding branches remain flat below $\approx$ 10 K.

In principle, six spin wave branches are expected in this model. However, since $K_1\approx K_2$, these branches should be considered as three almost doubly degenerate modes. Moreover, due to the hexagonal symmetry, these three branches can be described by a single one, stemming from $(0,0,\ell )$ and going soft at $(3,0,\ell )$. The two other branches are identical but stem from $(1,0,\ell )$ and $(2,0,\ell )$ and go soft respectively at $(4,0,\ell )$ and $(5,0,\ell )$. As shown in Fig.2(c,d), spin wave calculations indicate that the former corresponds mostly to correlations between spin components along the c-axis (Fig.2c), while the two latter correspond mostly to in-plane correlations (Fig.2d). As a result, two acoustic-like and one optical-like spin wave emerge from each zone center. The gap $\Delta$ of the latter occurs at around 40 cm${}^{-1}$ (5 meV) and is related to the easy-plane anisotropy by $\Delta \approx 3S\sqrt{D_{z}J}$.

As explained above, the degeneracy is lifted as soon as $K_1\ne K_2$. This becomes especially clear when looking at the dispersion along $(1,0,\ell )$. This point is illustrated in Fig.2e-f, showing the calculated spin wave dispersions for the different ${\Gamma }_i$ magnetic configurations. We notice that the spectral weight of the lowest energy mode is strong at $(1,0,1)$ for ${\Gamma }_{1,3}$, and at $(1,0,0)$ in the case of ${\Gamma }_{2,4}$. This change has been put forward in Ref. Fabreges2009, to explain that the spin reorientation process in HoMnO${}_3$ involves a symmetry change from ${\Gamma }_{2,4}$ below T${}_N$ to ${\Gamma }_{1,3}$ below T${}_{SR}$ with a 180${}^{\circ }$ rotation of the spins in one of the triangular layer. This change was attributed to a change of sign of $K_1-K_2$ induced by the change in position of the Mn ions from $x_{Mn}=0.325$ above T${}_{SR}$ to $x_{Mn}\approx 0.334$ below. In Ref. Fabreges2009, , the authors assumed ${\Gamma }_1$ to be the ground state. Actually, non linear optical measurements as well as other probes suggest rather a ${\Gamma }_3$ magnetic order below T${}_{SR}$ Fiebig2002 (); Brown2006 (); Nandi2008 (). Since the ${\Gamma }_{1,3}$ and ${\Gamma }_{2,4}$ pairs are homometric, i.e. they cannot be distinguished by unpolarized neutron scattering if $x_{Mn}$ is strictly equal to 1/3, we will assume that ${\Gamma }_3$ is the actual ground state in the following.

With these results in hand, we come back to the analysis of the experimental results. Fig.2b displays the calculated inelastic neutron intensity for Q wave vectors along $(h,0,1)$. Using $J=$2.5 meV and $D_z=$0.65 meV, the high energy behavior of the spectrum is well reproduced. For instance, the spin wave labeled M originates from the optical-like mode at $\Delta$. It is clearly visible on Figure 2. In Fig.3 (panel A), it looks like a flat band along $c^{\ast }$ due to the weakness of the interplane couplings $K_1$ and $K_2$. Finally, as shown in panel B, $\Delta$ hardens towards 45 cm${}^{-1}$ when decreasing temperature. Note that some renormalisation of the Mn excitations has been reported in HoMnO${}_3$ as due to magnon-phonon coupling Kim2018 ().

The second dispersive spin wave labeled M’ is the acoustic-like spin wave mode stemming from the zone center and consisting of two interlaced branches. It can be also identified in Fig.3, thanks to the strong intensity expected for ${\Gamma }_{2,4}$ structures at (1, 0, 0). Below T${}_{SR}$, this spectral weight jumps to (1, 0, 1), as a result of the transition to ${\Gamma }_{1,3}$. From panel B in Figure 3, the change of spectral weight from (1,0,0) to (1,0,1) at T${}_{SR}$ can also be noticed. The bandwidth of M’ is well captured for $K_1-K_2=0.003$ meV. The gap of this mode originates from small single-ion anisotropies of the Mn${}^{3+}$ (terms $D_y$ and $D_x$). An additional origin of anisotropy is expected to come from the Ho molecular field that should lead to a scaling of the gap with the Ho(4b) ordered moment. Indeed, at the mean field level, the coupling between the 3d and 4f magnetic species writes:

where $g$ is the coupling tensor, $m,p$ and $i,\ell$ are indices that label the unit cell and the atom for the 3d and rare earth sublattices. $⟨J_{p,\ell }⟩$ is the average magnetic moment at the rare earth site. Combined with the exchange Hamiltonian described in Eq. 1, $V$ polarizes the Mn spins, hence plays the role of anisotropy. In principle, this should result in a gap in the spin wave spectrum as observed in YbMnO${}_3$ Fabreges2010 (). However, in the present case, the expected scaling seems not to fully operate probably due to the coupling of the Mn spin waves with Ho CF excitations.

Indeed, in addition to the spin waves, non-dispersive excitations are also visible in Fig.2 and 3. These features correspond to CF transitions of the Ho${}^{3+}$ ions on 4b and 2a sites and are labeled 0-6. As we shall see in the next part, those labels are also used to describe the THz data (see Figure 4). Note that the mode 5 is not visible in our neutron scattering experiments probably because it overlaps with the magnon M or because its neutron cross section is intrinsically weak. Panel B of Figure 3 shows the temperature dependence of CF transition modes 1 to 4. The intensity of those modes gradually increases with decreasing temperature while their positions slightly change, but the most remarkable feature is the formation of an additional line labeled 1 just below T${}_{SR}$. Note that the line labeled 0 appears at very low temperature only, hence probably due to the Ho ordering on site 2a. In addition, quite strikingly, the dispersion of M’ eventually merges gradually below T${}_{SR}$ into those CF lines (at least into those below 30 cm${}^{-1}$). Finally, the spectrum encompasses only flat bands below $\approx$ 10 K.

We point out that this very peculiar temperature evolution of M’ shows that this branch crosses several Ho crystal field excitations, which makes very likely hybridization mechanisms (via a resonant process) between the Mn spin wave dispersion and the CF lines at low temperature.

To get a better insight into the spin dynamics, we have used THz spectroscopy as a complementary probe of both Mn spin waves and Ho Crystal field excitations. In this electromagnetic absorption technique, due to the high velocity of photons in the material, the measurements are considered in the long wave length limit and comparison with neutron data is therefore relevant for the (0, 0, 0) center of the Brillouin zone. There, all Ho crystal field excitations should be present. As for the the Mn spin waves, calculations similar to those presented in the previous section show that the four different spin configurations ${\Gamma }_i$ have the same signature at (0,0,0). It consists of a high energy branch (M) expected around 45 cm${}^{-1}$ which involves correlations between spin components perpendicular to the c-axis, and a single low energy branch M’ corresponding to spin correlations along the c-axis that is pushed to finite energy when some spin anisotropy is present (see above and Fig.2).

The THz spectrum of HoMnO${}_3$ was first studied in Talbayev2008 () where the Mn (M) magnon mode as well as several Ho crystal field excitations were observed as a function of external magnetic field. The Mn magnon was more recently investigated under light irradiation Bowlan2016 () while its dependence on the Ho-Mn interaction is reported in Laurita2017 (). Here, thanks to synchrotron based spectroscopy, we are able to measure with great precision the whole spectra from 0.3 to 1.8 THz (10 to 60 cm${}^{-1}$).

All the spectra recorded at different temperatures from 130 K or 100 K down to 6 K are shown in Fig.4 for the three possible orientations (as regards the c-axis) of the THz electric e and magnetic h fields. Several absorption bands are observed from 11 cm${}^{-1}$ up to 55 cm${}^{-1}$, quite differently from other hexagonal manganites such as YMnO${}_3$ or ErMnO${}_3$, suggesting the influence of Ho ions in these THz spectra. Further inspection shows that, for the two different THz polarizations with the same magnetic field direction h$\perp$c, the spectra are very similar and differ substantially from the case where $h\parallel c$. This is a strong indication that all the observed excitations are of magnetic origin. The Mn magnon (M) expected around 45 cm${}^{-1}$ is clearly observed for h$\perp$c and seems to behave as expected for a conventional magnon by emerging below T${}_N$. It shifts to higher energies when the temperature is lowered (see Fig.5). This magnon involves spin components perpendicular to this axis, in agreement with the THz magnetic selection rule h$\perp$c. The other Mn magnon (M’) involving the spin component along the c-axis is expected at lower energies for $h\parallel c$ but is not easily distinguishable. All the other observed bands are specific to the Ho compound and should be related to Ho crystal field excitations. Quite surprisingly, all spectra are strongly affected by the spin reorientation process at T${}_{SR}$.

A detailed description of the temperature dependence of the spectra are given in Fig.5 to Fig.9. Six different bands (each possibly including several excitations) are clearly visible together with the Mn magnon. The absorption peaks have been fitted with gaussian profile and their position, spectral weight (area) and lifetime (width) are reported. They all show, at the reorientation transition T${}_{SR}$, a slight change of their position in energy and more or less pronounced change of spectral weight. In particular band 1 is observable only below T${}_{SR}$ and for h$\perp$c (see Fig.6). Band 2, always present for h$\perp$c, is observable only below T${}_{SR}$ when $h\parallel c$ (see Fig.6). Band 3, always present for h$\perp$c, disappears below T${}_{SR}$ when $h\parallel c$, while band 4 has a strong increase below T${}_{SR}$ (see Fig.7). Note that band 3 below T${}_{SR}$ might have a shoulder and that band 4 encompasses certainly several modes. Finally, band 6 and maybe also band 5 have a side band that increases below T${}_{SR}$ (see Figs.8 and 9). The Mn magnon (M) is also affected by the spin reorientation at T${}_{SR}$: an abrupt change, although weak, occurs in its position and line width while its area, which was increasing below T${}_N$, starts to decrease and becomes temperature independent below 10 K (see Fig.5).

A comparison with the neutron data can be made keeping in mind that neutron scattering measures the imaginary part of the dynamical susceptibility that is equal to the THz absorption divided by the energy. Both techniques show remarkable agreement: At 20 K, below T${}_{SR}$, the observed five flat bands along $(1,0,\ell )$) in the neutron data (see Fig.3) correspond nicely to band 1 - 4 and to the magnon M observed in THz spectroscopy. The band 6 is also visible as a non dispersive branch at low temperature in the neutron data of Fig.2. At 45 K, above T${}_{SR}$, five branches are visible in the neutron spectra, three of them correspond to bands 2-3-4 in THz, while the dispersive magnon M’ is not distinguishable in THz, probably because it is below the energy window of the THz spectroscopy or mixed inside bands 3-4 (see Fig.3). To go further into the analysis of the different magnon/crystal field contributions and their possible hybridization, we propose additional modeling of the complex Mn and Ho dynamics. We first focus on the crystal field excitations associated to the Ho ions.

To go further, we took into account the coupling of the Ho 4b magnetic moments with the Mn ones expected to occur below T${}_{SR}$ (discarding a possible magnetic coupling of the Ho 2a with Mn) Nandi2008 (). This was achieved by including in the calculations the influence of the local magnetic field produced by the Mn moments on the Ho 4b site, similarly to the case of ErMnO${}_3$ Chaix2014 (). This field was chosen along the c-axis according to the observed c-direction of the Ho ordered magnetic moments. Its value was adjusted at 2 T so that the ordered Ho 4b magnetic moment extracted from the neutron diffraction data is reproduced at low temperature (see Appendix I). The influence of this magnetic field is to change slightly the energy of some peaks and to produce additional splitting of some CF levels (see Fig.11b and c for the parameter set 1). We can notice in particular that the lowest energy CF peak corresponding to Ho 4b (in blue) in zero field splits into two peaks under magnetic field. However, experimentally, there are abrupt changes at T${}_{SR}$ for all the THz bands, both on their position and their intensity, which cannot be accounted for by the sole effects of temperature and molecular field. This is a strong indication that electric effects with modification of electric charge screening are at play at T${}_{SR}$. This is further corroborated by the observation simultaneously of a sharp anomaly in the dielectric constant as well as a jump in the electric polarisation along the c-axis Hur2009 ().

With little additional changes of some CF parameters (set 2 in Table 2), we can reproduce qualitatively most of the observed features in the THz spectra (see Fig. 13d for T=6 K and a field of 2 T). In particular, bands 2, 3, 4, 5 and 6 are predicted at energies closer to the measured ones, although there are still some discrepancies with the calculated intensities. The presence of peak 3 above T${}_{SR}$ for $h\parallel c$ and its disappearance below T${}_{SR}$ while a new peak labeled 2 appears at lower energy could be due to the field splitting ascribed to the Ho-Mn coupling. Band 4 for $h\parallel c$ below T${}_{SR}$ could then result from a mixture of several peaks including the higher energy mode from the split peak (see horizontal bracket in Fig. 13d).

Strikingly, for $h\perp c$, band 1 appearing at T${}_{SR}$ in the measurements is missing in the calculations and therefore cannot be accounted for by the effect of a magnetic field. This discrepancy between calculations and experiment is another indication that hybridization is at play between Ho CF excitations and the lowest energy Mn magnon (M’).

Our analysis and calculations for the Mn magnetic order and associated spin waves, using the spin Hamiltonian of eq.1, can account for most of the experimental data. Mainly, two spin branches M and M’ are observed with INS that are correctly calculated (Fig.2 and 3). In particular, the spin reorientation at T${}_{SR}$ is clearly seen as a change in the spin wave dispersion along $c^{\ast }$ for the branch M’ and is correctly captured when the change of Mn spin configuration from ${\Gamma }_4$ to ${\Gamma }_3$ is introduced. The second branch M is also observed with THz spectroscopy (Fig.5), in agreement with the model. As for the non dispersive Ho CF excitations observed with INS and THz spectroscopy, our calculations using the CF hamiltonian described by eq.4 and 5 is able to reproduce the experimental data qualitatively when two ingredients are included: first, a change in charge screening at T${}_{SR}$ that explains the observed shifts and changes in intensity of the whole CF spectrum, and second, a molecular field along the c-axis produced by the Mn order. Similarly, a molecular field produced by the Ho 4b order may be introduced to account for the increased anisotropy gap in the Mn spin wave branch M’. However our calculations do not agree quantitatively with the whole experimental data: they cannot describe properly how the Mn spin branch M’ become non dispersive along $(1,0,\ell )$ and seems to merge into Ho CF excitations below 20 K. Moreover, they cannot explain the THz band 1 that appears at T${}_{SR}$ and is locked at 3 cm${}^{-1}$ below band 2 (Fig.6). Additional ingredients should be at play there, that couple the 3d and 4f magnetic species dynamically.

It is possible for Mn Magnons and Ho crystal field excitations to hybridize, if they are close in energy, which is the case for the Ho manganite where 6 different Ho crystal field excitations lie in the energy range of the Mn spin waves (see Fig.LABEL:INS2-V4). They also should not be forbidden by symmetry. This can be checked using a group representation analysis for both the Mn spin waves and the Ho crystal field excitations : if they contain at least one common irreducible representation, the coupling is allowed. The details are given in appendix II. According to Sikora1988 (), for the Mn magnetic orders with zero propagation vector corresponding to the four different one dimensionnal ${\Gamma }_i$, the spin wave representation may be decomposed into $\tau =A_2⨁B_1⨁E{}_1⨁E{}_2$ where we use the irreducible representations of the C${}_{6v}$ (6mm) point group. For Ho 4b and 2a crystal field excitations, their decompositions are already given in Fig.10, in the local point groups associated to the two different sites: C${}_3$ (3) for Ho 4b, and C${}_{3v}$ (3m) for Ho 2a. Their energy scheme contains singlets ($A$ for Ho 4b, $A_1$, $A_2$ for Ho 2a) and doublets ($E_1$, $E_2$ for Ho 4b, $E$ for Ho 2a). To infer compatibility with those of the Mn magnons, we must consider their decomposition into the irreducible representations of the C${}_{6v}$ point group. For Ho 4b, singlets $A$ are decomposed into {$A_1,A_2,B_1,B_2$} and degenerate doublets ($E_1⨁E{}_2$ are decomposed into {$E_1,E{}_2$}. As for the Ho 2a, singlets $A_1$ and $A_2$ are decomposed into {$A_1,B_2$} and {$A_2,B_1$} respectively, and doublets $E$ into {$E_1,E{}_2$} as for Ho 4b. It can be easily seen that they all have in common at least one irreducible representation contained in the Mn spin wave representation except for Ho 2a $A_1$ singlet that is calculated around 40-45 cm${}^{-1}$ and corresponds to band 5.

We conclude then that, in the energy range relevant for the experimental data reported in this study, all the CF excitations, except one (band 5 associated to Ho 2a), are allowed to couple to Mn spin waves. Looking back at neutron experiments and the spin wave calculations reproduced in Fig.12, it can be seen that, when the temperature is lowered close to T${}_{SR}$, the magnon M’ with small gaps at the zone centers (1,0,0) and (1,0,1) become close in energy to Ho (4b) CF excitations (bands 2 and 3-4), with different spectral weight for the magnons depending on the magnetic order ${\Gamma }_{1,3}$ or ${\Gamma }_{2,4}$. A coupling mechanism may be at play here that contributes to the spin reorientation process and the lost of dispersion for M’ along $c^{\ast }$.

It is worth noting that Ho and Mn ordered magnetic moments below T${}_{SR}$ are orthogonal. This suggests that, in the Mn - Ho coupling Hamiltonian of eq.III, the coupling tensor $g$ shows non diagonal terms, as it is the case for instance for the pseudo dipolar, the Dzyaloshinkii-Moriya interaction Fabreges2010 () or the highest order trigonal anisotropy Plumer2010 (). Furthermore, the 2 T effective magnetic field experienced by the 4b Ho suggests a coupling strength $g$ $\sim$ 0.1 K. In this context and given the neutron scattering data and the CF analysis detailed above, we may to consider a possible dynamical coupling between i) in plane Mn spin wave modes with Ho CF transitions polarized along the c-axis and ii) out of plane Mn spin wave modes with Ho CF transitions polarized in the (a,b) plane. In this view, it is tempting to assign band 1 to the Mn spin wave hybridized with band 2 (and also likely with band 3). In this scenario, the in plane character of the CF transitions would be transferred to the hybrid mode which would now appear in the h$\perp$c channel as observed experimentally in the THz measurements. This hybridisation occurs only below T${}_{SR}$ when the spectral weight of these excitations is enhanced and their energy matches. The energy difference between band 1 and 2, around 3 cm${}^{-1}$ experimentally, is in agreement with the coupling strength $g$ when all the Ho-Mn links are considered (12 of them).

To summarize, we have reexamined the THz dynamics present in the multiferroic compound HoMnO${}_3$. The peculiarity of this Ho compound in the hexagonal manganites is the occurrence of a spin reorientation below T${}_{SR}$=37 K. Remarkably, several Ho CF excitations are present in the same energy range as the Mn spin waves. Thanks to the complementary neutron and electromagnetic wave probes, we have identified five Ho 4b and Ho 2a CF excitations and shown how Mn spin waves as well as Ho CF are affected by the reorientation process at T${}_{SR}$. Using spin wave and CF calculations, we are able to reproduce qualitatively most of the observed features. At T${}_{SR}$, the Mn spin configuration switches from the high temperature ${\Gamma }_4$ configuration to ${\Gamma }_3$. A change in the spectral weight is observed along the $(1,0,\ell )$ direction that is correctly calculated in a simple hamiltonian for the Mn win waves. At the same time, changes occur in the CF excitations that can be accounted for when coupling to the lattice and surrounding oxygen charges are introduced. Then, Ho CF excitations start to match in energy the Mn branch at different Brillouin zone centers. Further inspection of the neutron and THz data points towards additional Ho-Mn coupling mechanisms that show up below T${}_{SR}$ in two different ways : first as an additional excitation observed in THz spectroscopy that is locked 3 cm${}^{-1}$ below a CF excitation and secondly the progressive disappearance of the Mn dispersion along $c^{\ast }$. Our detailed study on HoMnO${}_3$ gives finally new insight on the very rich rare earth - transition metal dynamical couplings that show up in the THz range.