Hilbert's 16th Problem for Quadratic Systems. New Methods Based on a Transformation to the Lienard Equation

Fractionally-quadratic transformations which reduce any two-dimensional quadratic system to the special Lienard equation are introduced. Existence criteria of cycles are obtained

Skyrmion robustness in non-centrosymmetric magnets with axial symmetry: The role of anisotropy and tilted magnetic fields

We investigate the stability of N\'eel skyrmions against tilted magnetic fields, in polar magnets with uniaxial anisotropy ranging from easy-plane to easy-axis type. We construct the corresponding phase diagrams and investigate the internal structure of skewed skyrmions with displaced cores. We find that moderate easy-plane anisotropy increases the stability range of N\'eel skyrmions for fields along the symmetry axis, while moderate easy-axis anisotropy enhances their robustness against tilted magnetic fields. We stress that the direction, along which the skyrmion cores are shifted, depends on the symmetry of the underlying crystal lattice. The cores of N\'eel skyrmions, realized in polar magnets with C$_{nv}$ symmetry, are displaced either along or opposite to the off-axis (in-plane) component of the magnetic field depending on the rotation sense of the magnetization, dictated by the sign of the Dzyaloshinskii constant. The core shift of antiskyrmions, present in chiral magnets with D$_{2d}$ symmetry, depends on the in-plane orientation of the magnetic field and can be parallel, anti-parallel, or perpendicular to it. We argue that the role of anisotropy in magnets with axially symmetric crystal structure is different from that in cubic helimagnets. Our results can be applied to address recent experiments on polar magnets with C$_{3v}$ symmetry, GaV$_4$S$_8$ and GaV$_4$Se$_8$

Asymmetric isolated skyrmions in polar magnets with easy-plane anisotropy

We introduce a new class of isolated magnetic skyrmions emerging within tilted ferromagnetic phases of polar magnets with easy-plane anisotropy. The asymmetric magnetic structure of these skyrmions is associated with an intricate pattern of the energy density, which exhibits positive and negative asymptotics with respect to the surrounding state with a ferromagnetic moment tilted away from the polar axis. Correspondingly, the skyrmion-skyrmion interaction has an anisotropic character and can be either attractive or repulsive depending on the relative orientation of the skyrmion pair. We investigate the stability of these novel asymmetric skyrmions against the elliptical cone state and follow their transformation into axisymmetric skyrmions, when the tilted ferromagnetic moment of the host phase is reduced. Our theory gives clear directions for experimental studies of isolated asymmetric skyrmions and their clusters embedded in tilted ferromagnetic phases

Magnetic Collapse and the Behavior of Transition Metal Oxides at High Pressure

We report a detail theoretical study of the electronic structure and phase stability of transition metal oxides MnO, FeO, CoO, and NiO in their paramagnetic cubic B1 structure by employing dynamical mean-field theory of correlated electrons combined with \emph{ab initio} band structure methods (DFT+DMFT). Our calculations reveal that under pressure these materials exhibit a Mott insulator-metal transition (IMT) which is accompanied by a simultaneous collapse of local magnetic moments and lattice volume, implying a complex interplay between chemical bonding and electronic correlations. Moreover, our results for the transition pressure show a monotonous decrease from ~ 145 GPa to 40 GPa, upon moving from MnO to CoO. In contrast to that, in NiO, magnetic collapse is found to occur at remarkably higher pressure of ~ 429 GPa. We provide a unified picture of such a behavior and suggest that it is primary a localized to itinerant moment behavior transition at the IMT that gives rise to magnetic collapse in transition metal oxides.Comment: 6 pages, 3 figure

Pressure-induced spin-state transition of iron in magnesiow\"ustite (Fe,Mg)O

We present a detailed theoretical study of the electronic, magnetic, and structural properties of magnesiow\"ustite Fe$_{1-x}$Mg$_x$O with $x$ in the range between 0$-$0.875 using a fully charge self-consistent implementation of the density functional theory plus dynamical mean-field theory (DFT+DMFT) method. In particular, we compute the electronic structure and phase stability of the rock-salt B1-structured (Fe,Mg)O at high pressures relevant for the Earth's lower mantle. We obtain that upon compression paramagnetic (Fe,Mg)O exhibits a spin-state transition of Fe$^{2+}$ ions from a high-spin to low-spin (HS-LS) state which is accompanied by a collapse of local magnetic moments. The HS-LS transition results in a substantial drop of the lattice volume by about 4$-$8 %, implying a complex interplay between electronic and lattice degrees of freedom. Our results reveal a strong sensitivity of the calculated transition pressure $P_{\rm tr.}$ upon addition of Mg. While for Fe-rich magnesiow\"ustite, Mg $x < 0.5$, $P_{\rm tr.}$ exhibits a rather weak variation at $\sim$80 GPa, for Fe-poor (Fe,Mg)O it drops, e.g., by about 35 % to 52 GPa for Mg $x=0.75$. This behavior is accompanied by a substantial change of the spin transition range from 50$-$140 GPa in FeO to 30$-$90 GPa for $x=0.75$. In addition, the calculated bulk modulus (in the HS state) is found to increase by $\sim$12 % from 142 GPa in FeO to 159 GPa in (Fe,Mg)O with Mg $x=0.875$. We find that the pressure-induced HS-LS transition has different consequences for the electronic properties of the Fe-rich and poor (Fe,Mg)O. For the Fe-rich (Fe,Mg)O, the transition is found to be accompanied by a Mott insulator to (semi-) metal phase transition. In contrast to that, for $x>0.25$, (Fe,Mg)O remains insulating up to the highest studied pressures, implying a Mott insulator to band insulator phase transition at the HS-LS transformation.Comment: 9 pages, 9 figure

Electronic structure of charge-ordered Fe3O4 from calculated optical, megneto-optical Kerr effect, and O K-edge x-ray absorption spectra

The electronic structure of the low-temperature (LT) monoclinic magnetite, Fe3O4, is investigated using the local spin density approximation (LSDA) and the LSDA+U method. The self-consistent charge ordered LSDA+U solution has a pronounced [001] charge density wave character. In addition, a minor [00{1/2}] modulation in the phase of the charge order (CO) also occurs. While the existence of CO is evidenced by the large difference between the occupancies of the minority spin t_{2g} states of ``2+'' and ``3+'' Fe_B cations, the total 3d charge disproportion is small, in accord with the valence-bond-sum analysis of structural data. Weak Fe orbital moments of ~0.07 mB are obtained from relativistic calculations for the CO phase which is in good agreement with recent x-ray magnetic circular dichroism measurements. Optical, magneto-optical Kerr effect, and O K-edge x-ray absorption spectra calculated for the charge ordered LSDA+U solution are compared to corresponding LSDA spectra and to available experimental data. Reasonably good agreement between the theoretical and experimental spectra supports the relevance of the CO solution obtained for the monoclinic LT phase. The results of calculations of effective exchange coupling constants between Fe spin magnetic moments are also presented.Comment: 32 pages, 10 figure