Quantum fluctuations of charge and phase transitions of a large Coulomb-blockaded quantum dot

We analyze ground-state properties of a large gated quantum dot coupled via a quantum point contact to a reservoir of one-dimensional spinless electrons whose interactions are characterized by the Luttinger liquid parameter g. We find that the classical steplike dependence of the average number of electrons on the dot n as a function of the gate voltage $n_x$ is preserved under certain conditions, the presence of quantum fluctuations notwithstanding. We point out that in its low-energy limit the problem is dual to that of a single-junction superconducting quantum interference device with Caldeira-Leggett dissipative coupling, and analogous to the classical problem of multilayer adsorption, and so is related to the classical statistical-mechanical problem of a one-dimensional Ising model with exchange interactions decaying as the inverse square of distance. This Ising universality class further subdivides into what we term (i) the Kondo/Ising class and (ii) the tricritical class. (i) For systems of the Kondo/Ising class, the ${n(n}_x)$ dependence is always continuous for $g>~1,$ while for $g<\mathrm{}1\mathrm{}$ (repulsive electrons in the constriction) the ${n(n}_x)$ dependence is continuous for sufficiently open dots, while taking the form of a modified staircase for dots sufficiently isolated from the reservoir. At the phase transition between the two regimes the magnitude of the dot population jump is only determined by the properties of the reservoir and is given by $g^{1/2}.$ (ii) For systems in the tricritical class we find in addition an intermediate regime where the dot population jumps from near-integer value to a region of stable population centered about a half-integer value. In particular, this tricritical behavior (together with the two dependences already seen in the Kondo/Ising class) is realized for noninteracting electrons, $g=1.\mathrm{}$