We report a two-particle semi-analytical approach towards a single quantum im-purity attached to two biased metallic leads. The theory is based on reduced Parquet equations, which are formulated near the critical point, giving a self-consistent way to treat the one-particle and two-particle quantities on the same footing.[1,2] The Kondo logarithmic scaling is analytically fulfilled at equilibrium in the strong interaction limit. Due to the renormalization of the two-particle vertices, the spurious transition towards magnetic order, which occurred in Hartree-Fock and GW approximations, is completely suppressed and the unphysical hysteresis loop in current-voltage characteristics is eliminated. Furthermore, in the linear re-sponse regime, we qualitatively reproduced three transport regimes with the in-creasing of the temperature: from Kondo resonant tunneling through Coulomb blockade regime and finally to sequential tunneling regime. In the case far away from equilibrium, we found the bias, in some senses, plays a similar role as the tem-perature which destroys the Kondo resonant peak when the corresponding ener-gy scale is comparable with the Kondo temperature. Besides, the applied voltage also develops local spectral peaks around each local chemical potentials in low bias, which agrees with the previous theoretical and experimental results.
If the time permits, I will also speak a little bit about our recent extension of this the-ory to the superconducting quantum dot. In our theory, due to the suppression of the spurious transition to magnetic order, the magnetic susceptibility in both the 0 and PI phases are qualitatively predicted. Furthermore, the increase of the tem-perature broadens the Andreev bound state, which can only be correctly repro-duced by a fully self-consistent theory.
 J. Yan and V. Janiš, arXiv:2109.11935
 V. Janiš, P. Zalom, V. Pokorný and A. Klíč, Phys. Rev. B 100, 195114 (2019)
 V. Janiš and J. Yan Phys. Rev. B 103, 235163 (2021)
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