## 1 Introduction  
Hole-doped colossal magnetoresistive (CMR) manganites with general formula RE\({}_{1-x}\)AE\({}_{x}\)MnO\({}_{3}\) (RE = trivalent rare-earth or La\({}^{3+}\), AE = divalent alkaline-earth metal Ca\({}^{2+}\), Sr\({}^{2+}\), Ba\({}^{2+}\), 0 \(\leq\) x \(\leq\) 1, have been the subject of intense research for over twenty years thanks to their rich physics. This emerges from an outstanding variety of interconnected structural, electronic, and magnetic phase transitions, which lead to competing ground states, either metallic ferromagnetic (FM), with predominant Zener double exchange interaction [1], or insulating antiferromagnetic (AFM), with predominant [PERSON] superexchange interaction [2]. The crystal structures of the undoped phases depend on the RE size, changing from orthorhombic GdFeO\({}_{3}\)-type perovskite (space group n. 62 _Pbnm_) for large rare-earths (from La to Tb), to hexagonal non-perovskite LuMnO\({}_{3}\)-type (space group n. 185 \(P\delta_{3}cm\)) for small ones (from Dy to Lu, and Y). The partial substitution of RE\({}^{3+}\) by a larger AE\({}^{2+}\) ion induces an increase of the tolerance factor t = \(\left[\left(r_{\rm A}\right)+r_{\rm O}\right]/\left[\left.\sqrt{2}\left(r_{\rm B }+r_{\rm O}\right)\right.\right]\) expressed, with reference to the ABO\({}_{3}\) simple perovskite chemical formula, in terms of octahedral B-site cation and oxygen radii, respectively, r\({}_{\rm B}\) and r\({}_{\rm O}\), and of the \(<\)r\({}_{\rm A}\)\(>\) radius of the A-type cation, this latter averaged between the two heterovalent A-type cations. As a consequence, the hexagonal structure is converted back to orthorhombic as long as the hole concentration exceeds a critical value x\({}_{\text{c}}\). In particular, all of the Ca-substituted half-doped phases RE\({}_{0.5}\)Ca\({}_{0.5}\)MnO\({}_{3}\) are distorted _Pbmm_ perovskites. The holmium-based system Ho\({}_{1-x}\)Ca\({}_{x}\)MnO\({}_{3}\) has some interesting peculiarities. HoMnO\({}_{3}\) belongs to the subgroup of REMnO\({}_{3}\) phases, which adopt the hexagonal _P6\({}_{3}\)mc_ structure. HoMnO\({}_{3}\) is a multiferroic crystal in which ferroelectricity, with Curie temperature T\({}_{\text{C}}\) as high as 875 K, is coupled to a frustrated AFM order with Neel temperature T\({}_{\text{N}}\) of 72 K [3]. Since the size of Ho\({}^{3+}\) is located just at the hexagonal and orthorhombic structural phase boundary, HoMnO\({}_{3}\) also exists as a metastable _Pbmm_ perovskite [4; 5], which retains ferroelectric properties, however, induced by the AFM ordering of the Mn ions occurring at T\({}_{\text{N}}\) about 40 K, with a much lower ferroelectric Curie point of 30 K. On hole doping, Ho\({}_{1-x}\)Ca\({}_{x}\)MnO\({}_{3}\) suddenly converts to the stable _Pbmm_ structure already at x = 0.1 [6]. Ferroelectricity still exists below T\({}_{\text{N}}\) at x = 0.2 [6]. Half-doped Ho\({}_{0.5}\)Ca\({}_{0.5}\)MnO\({}_{3}\) presents a highly stable insulating AFM ground state. Ferroelectricity and its interconnections with antiferromagnetism have not been investigated anymore, outclassed in interest by the two phenomena of orbital ordering (OO) and charge ordering (CO), common to all Ca\({}^{2+}\) half-doped RE\({}_{1-x}\)Ca\({}_{x}\)MnO\({}_{3}\) manganites. OO and CO are two of the most fascinating transitions, structural and electronic, exhibited by CMR manganites. An introduction to the subject can be found in [7]. OO derives from the presence of Jahn-Teller (JT) active Mn\({}^{3+}\) (t\({}_{\text{2g}}\)\({}^{3}\)e\({}_{\text{8}}^{1}\)) ions at the B octahedral site of the ABO\({}_{3}\) perovskite, the simple cubic cell of which is shown in Figure 1a, evidencing the vertex-by-vertex octahedra interconnections. OO consists in a cooperative distortion of the whole crystal structure at a critical temperature T\({}_{\text{OO}}\), which reflects the occurrence of a static long-range ordered pattern of the 3d e\({}_{\text{g}}\) x\({}^{2}-\) y\({}^{2}\) or 3r\({}^{2}-\) z\({}^{2}\) orbitals.