A Solvable Regime of Disorder and Interactions in Ballistic Nanostructures, Part I: Consequences for Coulomb Blockade

Abstract

We provide a framework for analyzing the problem of interacting electrons in a ballistic quantum dot with chaotic boundary conditions within an energy $E_T$ (the Thouless energy) of the Fermi energy. Within this window we show that the interactions can be characterized by Landau Fermi liquid parameters. When $g$, the dimensionless conductance of the dot, is large, we find that 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). The infinite $g$ theory shows a transition to a strong-coupling phase characterized by the same order parameter as in the Pomeranchuk transition in clean systems (a spontaneous interaction-induced Fermi surface distortion), but smeared and pinned by disorder. At finite $g$, the two phases and critical point evolve into three regimes in the $u_m-1/g$ plane -- weak- and strong-coupling regimes separated by crossover lines from a quantum-critical regime controlled by the quantum critical point. In the strong-coupling and quantum-critical regions, the quasiparticle acquires a width of the same order as the level spacing $\Delta$ within a few $\Delta$'s of the Fermi energy due to coupling to collective excitations. In the strong coupling regime if $m$ is odd, the dot will (if isolated) cross over from the orthogonal to unitary ensemble for an exponentially small external flux, or will (if strongly coupled to leads) break time-reversal symmetry spontaneously.

33 pages, 14 figures. Very minor changes. We have clarified that we are treating charge-channel instabilities in spinful systems, leaving spin-channel instabilities for future work. No substantive results are changed

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• A Universal Interacting Crossover Regime in Two-Dimensional Quantum Dots
• Quantum criticality near the Stoner transition in a two-dot with spin-orbit coupling
• The twilight zone in the parametric evolution of eigenstates: beyond perturbation theory and semiclassics
• Interactions, superconducting $T_c$, and fluctuation magnetization for two coupled dots in the crossover between the Gaussian Orthogonal and Unitary ensembles
• Ballistic Quantum Dots with Disorder and Interactions: A numerical study on the Robnik-Berry billiard
• Diamagnetic persistent currents for electrons in ballistic billiards subject to a point flux
• Ballistic dynamics of a convex smooth-wall billiard with finite escape rate along the boundary
• Linear Magnetic Response of Disordered Metallic Rings: Large Contribution from Forward Scattering Interactions
• The renormalization group for interacting fermions: from Fermi liquids to quantum dots
• A nearly closed ballistic billiard with random boundary transmission