Non-equilibrium current and relaxation dynamics of a charge-fluctuating quantum dot

Abstract

We study the steady-state current in a minimal model for a quantum dot dominated by charge fluctuations and analytically describe the time evolution into this state. The current is driven by a finite bias voltage V across the dot, and two different renormalization group methods are used to treat small to intermediate local Coulomb interactions. The corresponding flow equations can be solved analytically which allows to identify all microscopic cutoff scales. Exploring the entire parameter space we find rich non-equilibrium physics which cannot be understood by simply considering the bias voltage as an infrared cutoff. For the experimentally relevant case of left-right asymmetric couplings, the current generically shows a power-law suppression for large V . The relaxation dynamics towards the steady state features characteristic oscillations as well as an interplay of exponential and power-law decay.

Functional renormalization group approach to correlated fermion systems

“…(166) It turns out to be useful (Andergassen et al, 2011;Karrasch et al, 2010a) to introduce the two scales…”

Relaxation dynamics in correlated quantum dots

“…For this reason, perturbative renormalization group (RG) approaches [2,3] have been introduced in the regime of weak coupling between dot and reservoirs. Using a recently developed real-time RG approach based on a diagrammatic expansion in Liouville space [2], we illustrate recent results for the nonlinear transport and the time evolution into the stationary state for two basic model systems: the interacting resonant level model (IRLM) describing a quantum dot dominated by charge fluctuations [4], and the Kondo model for a dot with spin fluctuations [5,6]. In the stationary state, the finite bias voltage V introduces new effects in the current due to the availability of additional transport channels.…”

“…Non-Markovian contributions arising from the z-dependence of the Liouvillian give rise to branch cuts. For a constant density of states in the leads (normal metallic case) it can be shown generically [4,6] that the poles lead to exponential decay with oscillations, whereas the branch cuts induce additional power-law behavior. This result is of particular importance for applications in error correction schemes [1].…”

“…The electrons in the reservoirs are assumed to be (effectively) non-interacting and held at two different chemical potentials μ α = ±V /2, with α = L, R denoting the left and right reservoir. By analytically solving the RG equations, we obtain closed integral representations both for the dot occupation n(t) and the current I(t) by performing an inverse Laplace transform [4]. The characteristics of the time evolution of both n(t) as well as the current I(t) are that (i) the relaxation towards the stationary value is governed by two decay rates Γ 1 and Γ 2 , where Γ 1 describes the charge relaxation and Γ 2 is the broadening of the local level induced by the coupling to the reservoirs; (ii) the voltage appears as an important energy scale for the dynamics setting the frequency of an oscillatory behavior; and (iii) the exponential decay is accompanied by an algebraic decay ∼ t 2U−1 (see Fig.…”

“…Time evolution of the dot occupation n(t) and the current I(t) for the IRLM with U = 0.1/π and ε = 10T K [4]. T K denotes the Kondo scale used as unit of energy.…”