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 interacting spinless electrons. We find that the classical step-like dependence of the dot population on the gate voltage is preserved under certain conditions. We point out that the problem is related to the classical one-dimensional Ising model with inverse-square interactions. This Ising universality class further subdivides into (i) the Kondo/Ising class and (ii) the tricritical class. For systems of the Kondo/Ising class, and repulsive electrons, the gate voltage dependence of the dot population 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 universal. 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 behaviour (together with the two dependencies already seen in the Kondo/Ising class) is realized for non-interacting electrons. 
Eugene B. Kolomeisky, Robert M. Konik and Xiaoya Qi