Influence of electron-phonon interaction on quantum phase transition in the triple quantum dots coupled to pseudo-gap Fermi system

We investigate the triple quantum dots with triangular geometry, in which one of the dots (embedded dot) is coupled to pseudo-gap Fermi system and electrons in the dot interact with a local phonon mode. The model, which we treat using the analytical argument and the numerical renormalization group, exhibits the quantum phase transition between ferromagnetically-coupled local moment phase and antiferromagnetically-coupled strong coupling phase depending upon electron-phonon coupling. We restrict our discussion to the case that inter-dot exchange couplings are much smaller than the Kondo temperature for the embedded dot. Since Fermi system has the density of states dependent on energy in power-law fashion, the embedded dot retains an unscreened spin degree of freedom even in Kondo regime. For the coupling to such Fermi system, the simultaneous increase of electron-phonon coupling and pseudo-gap exponent has the contrary effects on the quantum phase transition depending upon inter-dot couplings.