Competition of crystal field splitting and Hund's rule coupling in two-orbital magnetic metal-insulator transitions

Competition of crystal field splitting and Hund's rule coupling in magnetic metal-insulator transitions of half-filled two-orbital Hubbard model is investigated by multi-orbital slave-boson mean field theory. We show that with the increase of Coulomb correlation, the system firstly transits from a paramagnetic (PM) metal to a {\it N\'{e}el} antiferromagnetic (AFM) Mott insulator, or a nonmagnetic orbital insulator, depending on the competition of crystal field splitting and the Hund's rule coupling. The different AFM Mott insulator, PM metal and orbital insulating phase are none, partially and fully orbital polarized, respectively. For a small $J_{H}$ and a finite crystal field, the orbital insulator is robust. Although the system is nonmagnetic, the phase boundary of the orbital insulator transition obviously shifts to the small $U$ regime after the magnetic correlations is taken into account. These results demonstrate that large crystal field splitting favors the formation of the orbital insulating phase, while large Hund's rule coupling tends to destroy it, driving the low-spin to high-spin transition.Comment: 4 pages, 4 figure

Interaction of free-floating planets with a star-planet pair

The recent discovery of free-floating planets and their theoretical interpretation as celestial bodies, either condensed independently or ejected from parent stars in tight clusters, introduced an intriguing possibility. Namely, that some exoplanets are not condensed from the protoplanetary disk of their parent star. In this novel scenario a free-floating planet interacts with an already existing planetary system, created in a tight cluster, and is captured as a new planet. In the present work we study this interaction process by integrating trajectories of planet-sized bodies, which encounter a binary system consisting of a Jupiter-sized planet revolving around a Sun-like star. To simplify the problem we assume coplanar orbits for the bound and the free-floating planet and an initially parabolic orbit for the free-floating planet. By calculating the uncertainty exponent, a quantity that measures the dependence of the final state of the system on small changes of the initial conditions, we show that the interaction process is a fractal classical scattering. The uncertainty exponent is in the range (0.2-0.3) and is a decreasing function of time. In this way we see that the statistical approach we follow to tackle the problem is justified. The possible final outcomes of this interaction are only four, namely flyby, planet exchange, capture or disruption. We give the probability of each outcome as a function of the incoming planet's mass. We find that the probability of exchange or capture (in prograde as well as retrograde orbits and for very long times) is non-negligible, a fact that might explain the possible future observations of planetary systems with orbits that are either retrograde or tight and highly eccentric.Comment: 19 pages, 12 figure

Zero dynamics and stabilization for linear DAEs

We study linear differential-algebraic multi-input multi-output systems which are not necessarily regular and investigate the asymptotic stability of the zero dynamics and stabilizability. To this end, the concepts of autonomous zero dynamics, transmission zeros, right-invertibility, stabilizability in the behavioral sense and detectability in the behavioral sense are introduced and algebraic characterizations are derived. It is then proved, for the class of right-invertible systems with autonomous zero dynamics, that asymptotic stability of the zero dynamics is equivalent to three conditions: stabilizability in the behavioral sense, detectability in the behavioral sense, and the condition that all transmission zeros of the system are in the open left complex half-plane. Furthermore, for the same class, it is shown that we can achieve, by a compatible control in the behavioral sense, that the Lyapunov exponent of the interconnected system equals the Lyapunov exponent of the zero dynamics