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Antiferromagnetic and structural transitions in the superoxide KO2 from first principles: A 2p-electron system with spin-orbital-lattice coupling
KO2 exhibits concomitant antiferromagnetic (AFM) and structural transitions,
both of which originate from the open-shell 2p electrons of O
molecules. The structural transition is accompanied by the coherent tilting of
O molecular axes. The interplay among the spin-orbital-lattice
degrees of freedom in KO2 is investigated by employing the first-principles
electronic structure theory and the kinetic-exchange interaction scheme. We
have shown that the insulating nature of the high symmetry phase of KO2 at high
temperature (T) arises from the combined effect of the spin-orbit coupling and
the strong Coulomb correlation of O 2p electrons. In contrast, for the low
symmetry phase of KO2 at low T with the tilted O molecular axes, the
band gap and the orbital ordering are driven by the combined effects of the
crystal-field and the strong Coulomb correlation. We have verified that the
emergence of the O 2p ferro-orbital ordering is essential to achieve the
observed AFM structure for KO2
Continuity of the Explosive Percolation Transition
The explosive percolation problem on the complete graph is investigated via
extensive numerical simulations. We obtain the cluster-size distribution at the
moment when the cluster size heterogeneity becomes maximum. The distribution is
found to be well described by the power-law form with the decay exponent , followed by a hump. We then use the finite-size scaling method to
make all the distributions at various system sizes up to collapse
perfectly onto a scaling curve characterized solely by the single exponent
. We also observe that the instant of that collapse converges to a
well-defined percolation threshold from below as . Based on
these observations, we show that the explosive percolation transition in the
model should be continuous, contrary to the widely-spread belief of its
discontinuity.Comment: Some corrections during the revie
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