We propose a variational approach for computing the macroscopic entanglement in a many-body mixed state, based on entanglement witness operators, and compute the entanglement of formation (EoF), a mixed-state generalization of the entanglement entrop
y, in single- and two-channel Kondo systems at finite temperature. The thermal suppression of the EoF obeys power-law scaling at low temperature. The scaling exponent is halved from the single- to the two-channel system, which is attributed, using a bosonization method, to the non-Fermi liquid behavior of a Majorana fermion, a half of a complex fermion, emerging in the two-channel system. Moreover, the EoF characterizes the size and power-law tail of the Kondo screening cloud of the single-channel system.
We fabricated Pt/NiO/Pt capacitor structures with various bottom electrode thicknesses, $t_{BE}$, and investigated their resistance switching behaviors. The capacitors with $t_{BE} geq 50$ nm exhibited typical unipolar resistance memory switching, wh
ile those with $t_{BE} leq 30$ nm showed threshold switching. This interesting phenomenon can be explained in terms of the temperature-dependent stability of conducting filaments. In particular, the thinner $t_{BE}$ makes dissipation of Joule heat less efficient, so the filaments will be at a higher temperature and become less stable. This study demonstrates the importance of heat dissipation in resistance random access memory.
