Quantum Dots with Disorder and Interactions: A Solvable Large-[Formula presented] Limit

We analyze the problem of interacting electrons on a ballistic quantum dot with chaotic boundary conditions, where the effective interactions at low energies are characterized by Landau parameters. When the dimensionless conductance [Formula presented] of the dot is large, the disordered interacting problem can be solved in a saddle-point approximation which becomes exact as [Formula presented] (as in a large-[Formula presented] theory), leading to a phase transition in each Landau interaction channel. In the weak-coupling phase constant charging and exchange interactions dominate the low-energy physics, while the strong-coupling phase displays a spontaneous distortion of the Fermi surface, smeared out by disorder.