In a recent work, Le Hur has shown that dissipative coupling to gate electrodes may play an important role in a quantum box near its degeneracy point [K. Le Hur, Phys. Rev. Lett. 92, 196804 (2004)]: While quantum fluctuations of the charge of the dot tend to round Coulomb blockade charging steps of the box, strong enough dissipation suppresses these fluctuations and leads to the reappearance of sharp charging steps. In the present paper we study this quantum phase transition in detail using bosonization and numerical renormalization group methods in the limit of vanishing level spacing.
We study the transport properties of a quantum dot (QD) with highly
resistive gate electrodes, and show that the QD displays a quantum
phase transition analogous to the famous dissipative phase transition
first identified by S. Chakravarty [Phys. Rev. Lett. 49, 681
(1982)]; for a review see [A. J. Leggett et al.,
Rev. Mod. Phys. 59, 1 (1987)]. At temperature T=0, the charge on
the central island of a conventional QD changes smoothly as a function
of gate voltage, due to quantum fluctuations. However, for
sufficiently large gate resistance charge fluctuations on the island
can freeze out even at the degeneracy point, causing the charge on the
island to change in sharp steps as a function of gate voltage. For
Rg < RC the steps remain smeared out by quantum fluctuations. The Coulomb blockade peaks in conductance display anomalous scaling at intermediate temperatures, and at very low temperatures a sharp step develops in the QD conductance.