Quantum Dots with Disorder and Interactions: A Solvable Large-$g$ 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 $g$ of the dot is large, the disordered interacting problem can be solved in a saddle-point approximation which becomes exact as $g\to \infty$ (as in a large-$N$ 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.

Figure 1

Schematic showing the critical point at $u_m^{*}=-1/\log 2$ in the $g\to \infty$ limit on the $x$ axis. As $g$ decreases one moves in the $y$ direction and the critical point controls the behavior of all physical quantities in the quantum critical region.