Many-body delocalization transition and relaxation in a quantum dot

We revisit the problem of quantum localization of many-body states in a quantum dot and the associated problem of relaxation of an excited state in a finite correlated electron system. We determine the localization threshold for the eigenstates in Fock space. We argue that the localization-delocalization transition (which manifests itself, e.g., in the statistics of many-body energy levels) becomes sharp in the limit of a large dimensionless conductance (or, equivalently, in the limit of weak interaction). We also analyze the temporal relaxation of quantum states of various types (a “hot-electron state,” a “typical” many-body state, and a single-electron excitation added to a “thermal state”) with energies below, at, and above the transition.

Figure 1

Illustration of a decay process of $p=2$ quasiparticles into 6 quasiparticles. (a) and (b) Diagrams for Schrödinger perturbation theory with two different virtual states a and b, respectively. Red vertical lines indicate the sequence of many-body states. (c) Corresponding interfering paths in Fock space.

Figure 2

Schematic illustration of the saturation crossover in the return probability $P\left(t\right)$ on the log-log scale across the delocalization transition for a hot electron with $E/\mathrm{\Delta }=5,7,10,15,20,25,30,40,50,60$ (from top to bottom) and $g=25$. Blue lines correspond to the localized regime, green lines to the delocalized regime, and the red line to the localization threshold. The slope of all lines corresponds to the stretched exponential decay, Eq. (55). The saturation time is given by Eq. (56) in the localized regime and by Eq. (61) in the delocalized regime. The saturation value of $-lnP\left(T\right)$ is given by Eq. (25) in the localized regime and by Eq. (7) in the delocalized regime.