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We consider the problem of a resonant level coupled to the edge of a one-dimensional lead with interactions both within the lead and between the level and the lead. The lead is assumed to be in a metallic Luttinger liquid phase. Using an exact mapping of the problem onto a classical one-dimensional Coulomb-gas model of the Yuval-Anderson type, we show that the level occupation depends on the interactions only through the corresponding Fermi edge singularity exponent, and is equal to the magnetization of an anisotropic Kondo-model with appropriately chosen exchange couplings. We find, however, that the level local density of states depends in a different way on each type of interaction. Thus, it is predicted that a level coupled to an interacting lead without level-lead interaction has the same population as a level coupled to a non-interacting lead with level-lead interaction of the opposite sign, although the local density of states is different in each case. In addition, the combination of interactions of the same sign both in the lead and between the level and the lead, may cause the gate-voltage dependence of the occupation of the coupled level to be broader than the corresponding non-interacting case, despite having a pseudo-gap in the local density of states, or the opposite way around. The predictions regarding the level population were verified to a high degree of accuracy using the density matrix renormalization group algorithm. The results agree with classical Monte-Carlo simulations performed on the corresponding Coulomb-gas model. The latter method also enables to extract the local density of states. |
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