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We study a quantum phase transition which occurs in a system composed of twoimpurities (or quantum dots) each coupled to a different interacting(Luttinger-liquid) lead. While the impurities are coupled electrostatically,there is no tunneling between them. Using a mapping of this system onto a Kondomodel, we show analytically that the system undergoes aBerezinskii-Kosterlitz-Thouless quantum phase transition as function of theLuttinger liquid parameter in the leads and the dot-lead interaction. The phasewith low values of the Luttinger-liquid parameter is characterized by an abruptswitch of the population between the impurities as function of a common appliedgate voltage. However, this behavior is hard to verify numerically since onewould have to study extremely long systems. Interestingly though, at thetransition the entanglement entropy drops from a finite value of $\ln(2)$ tozero. The drop becomes sharp for infinite systems. One can employ finite sizescaling to extrapolate the transition point and the behavior in its vicinityfrom the behavior of the entanglement entropy in moderate size samples. Weemploy the density matrix renormalization group numerical procedure tocalculate the entanglement entropy of systems with lead lengths of up to 480sites. Using finite size scaling we extract the transition value and show it tobe in good agreement with the analytical prediction.

Entanglement entropy and quantum phase transitions in quantum dots coupled to Luttinger liquid wires