Solvable regime of disorder and interactions in ballistic nanostructures: Consequences for Coulomb blockade

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 Thouless number (also the dimensionless conductance when it has strong coupling to leads) of the dot, is large, we find that the disordered interacting problem can be solved in a saddle-point approximation which becomes exact as $\overrightarrow{g}\infty$ (as in a $\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}-N$ theory). The infinite g theory has two phases as a function of the Landau parameter $u_m$ in a channel with angular momentum m: A weak-coupling phase where constant charging and exchange interactions dominate the low-energy physics, as in previous “universal Hamiltonian” treatments, and 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. Thus, both interactions and disorder are crucial to the existence of these phases. 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 this, the first of a two part series, we focus on the consequences of this picture for Coulomb Blockade experiments. We employ analytical and numerical methods to predict the statistics of single-particle levels, Coulomb Blockade peak spacings, conductance peak heights, and quasiparticle widths. We show that 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 $\Delta \mathrm{’}\mathrm{s}$ of the Fermi energy due to coupling to collective excitations. In the strong-coupling regime if m is odd, the dot will (if isolated) crossover from the orthogonal to unitary ensemble for an exponentially small external flux or will (if strongly coupled to leads) break time-reversal symmetry spontaneously. For any m, the peak-spacing distribution becomes broader than expected in previous works and even has support at negative values, which in turn is correlated with small peak heights. Ballistic/chaotic quantum dots afford us unrivalled theoretical and experimental control over the problem of simultaneous disorder and interactions due to the $1/g$ expansion and our ability to vary disorder and interaction much more readily than in the bulk.