The anisotropy $\gamma$ of the superconducting state of high quality single crystals of ${\mathrm{MgB}}_2$ was determined, using torque magnetometry with two different methods. The anisotropy of the upper critical field was found to be temperature dependent, decreasing from $\gamma \simeq 6$ at 15 K to 2.8 at 35 K. Reversible torque data near $T_c$ reveal a field dependent anisotropy, increasing nearly linearly from $\gamma \simeq 2$ in zero field to 3.7 in 10 kOe. The unusual temperature dependence is a true bulk property and can be explained by nonlocal effects of anisotropic pairing and/or the $\overrightarrow{k}$ dependence of the effective mass tensor.

We investigated the series of temperature and field-driven transitions in ${\mathrm{LuFe}}_2{\mathrm{O}}_4$ by optical and Mössbauer spectroscopies, magnetization, and x-ray scattering in order to understand the interplay between charge, structure, and magnetism in this multiferroic material. We demonstrate that charge fluctuation has an onset well below the charge ordering transition, supporting the “order by fluctuation” mechanism for the development of charge order superstructure. Bragg splitting and large magneto-optical contrast suggest a low-temperature monoclinic distortion that can be driven by both temperature and magnetic field.

We present single crystal neutron diffraction measurements on multiferroic ${\mathrm{LuFe}}_2{\mathrm{O}}_4$. Magnetic reflections are observed below transitions at 240 and 175 K indicating that the magnetic interactions in ${\mathrm{LuFe}}_2{\mathrm{O}}_4$ are three-dimensional in character. The magnetic structure is refined as a ferrimagnetic spin configuration below the 240 K transition. Below 175 K a significant broadening of the magnetic peaks is observed along with the buildup of a diffuse component to the magnetic scattering.

X-ray scattering by multiferroic ${\mathrm{LuFe}}_2{\mathrm{O}}_4$ is reported. Below 320 K, superstructure reflections indicate an incommensurate charge order with propagation close to $(\frac{1}{3}\frac{1}{3}\frac{3}{2})$. The corresponding charge configuration, also found by electronic structure calculations as most stable, contains polar $\mathrm{Fe}/\mathrm{O}$ double layers with antiferroelectric stacking. Diffuse scattering at 360 K, with $(\frac{1}{3}\frac{1}{3}0)$ propagation, indicates ferroelectric short-range correlations between neighboring double layers. The temperature dependence of the incommensuration indicates that charge order and magnetism are coupled.

We present the refinement of the crystal structure of charge-ordered ${\mathrm{LuFe}}_2{\mathrm{O}}_4$, based on single-crystal x-ray diffraction data. The arrangement of the different Fe-valence states, determined with bond-valence-sum analysis, corresponds to a stacking of charged Fe bilayers, in contrast with the polar bilayers previously suggested. This arrangement is supported by an analysis of x-ray magnetic circular dichroism spectra, which also evidences a strong charge-spin coupling. The nonpolar bilayers are inconsistent with charge order based ferroelectricity.