Functional renormalization group study of the interacting resonant level model in and out of equilibrium

We investigate equilibrium and steady-state nonequilibrium transport properties of a spinless resonant level locally coupled to two conduction bands of width $\sim \Gamma$ via a Coulomb interaction $U$ and a hybridization $t^{\prime }$. In order to study the effects of finite bias voltages beyond linear response, a generalization of the functional renormalization group to Keldysh frequency space is employed. Being mostly unexplored in the context of quantum impurity systems out of equilibrium, we benchmark this method against recently published time-dependent density matrix renormalization group data. We thoroughly investigate the scaling limit $\Gamma \to \infty$ characterized by the appearance of power laws. Most importantly, at the particle-hole symmetric point the steady-state current decays like $J\sim V^{-{\alpha }_J}$ as a function of the bias voltage $V⪢t^{\prime }$, with an exponent ${\alpha }_J\left(U\right)$ that we calculate to leading order in the Coulomb interaction strength. In contrast, we do not observe a pure power-law (but more complex) current-voltage-relation if the energy $ϵ$ of the resonant level is pinned close to either one of the chemical potentials $\pm V/2$.

Figure 1

(Color online) Schematic presentation of the two-channel interacting resonant level model studied in this paper.

Figure 2

(Color online) The steady-state current $J$ as a function of the bias voltage $V$ of the two-channel IRLM for large hoppings $t^{\prime }=0.5\Gamma$ (in units of the bandwidth $\sim \Gamma$), zero impurity energy $ϵ$, and various Coulomb interactions $U$. (a) Functional renormalization group results obtained from numerical integration of the self-energy flow Eqs. (30) of the sharp cut-off scheme of Sec. 3C2. (b) The same calculated with the reservoir cut-off approach of Sec. 3C3. Density-matrix renormalization group data of Ref. 26 for the same set of parameters is shown by symbols in the main part (for $U=0.3\Gamma$ only) as well as within the inset (where lines are a guide to the eye only).

Figure 3

(Color online) Linear-response conductance (in units of $G_0=e^2/h$) of the IRLM at $t^{\prime }=0.1\Gamma$ as a function of the gate voltage $ϵ$. The figure shows a comparison between functional RG data obtained from the formalism of Sec. 3C1 (lines) and the DMRG results of Ref. 25 (symbols).

Figure 4

(Color online) The exponents ${\alpha }_J$ and ${\alpha }_{\chi }$ governing the scaling-limit power-law behavior of the current (for $t^{\prime }⪡V$) and of the susceptibility (for $V⪡t^{\prime }$), respectively. To leading order, both quantities are given by $\alpha =2U/\pi \Gamma$. For the equilibrium exponent ${\alpha }_{\chi }$, symbols show numerical renormalization group reference data.

Figure 5

(Color online) The current as a function of the voltage in the scaling limit $\Gamma \to \infty$ obtained numerically from the sharp (solid lines) and reservoir (dashed lines) FRG cut-off schemes, respectively. Beyond some cross-over scale $T_K$, $J$ decays as a power law of the voltage $V$ over several orders of magnitude. The latter manifests as a constant logarithmic derivative $d\text{ln}\left(J/\Gamma \right)/d\text{ln}\left(V/\Gamma \right)$.

Figure 6

(Color online) The same as Fig. 5, but for a finite impurity energy $ϵ$. The data was obtained by numerically integrating the flow Eqs. (36, 37, 38, 39). The current is suppressed for small voltages $V⪡ϵ$ but crosses over to a power-law decay when $ϵ$ is moved below the chemical potential. Inset: The quantity ${\alpha }_{\text{res}}$ governing the behavior close to the aforementioned resonance condition $ϵ\approx \pm V/2$ (see Eq. (52)).

Figure 7

(Color online) Upper panel: the same as in Fig. 3, but additionally showing FRG results obtained from the second order truncation scheme outlined in the Appendix. Lower panel: the logarithmic derivative $d\text{ln}\left(\Gamma /\chi t^{\prime 2}\right)/d\text{ln}\left(t^{\prime 2}/{\Gamma }^2\right)$ obtained from the FRG flow Eq. (25) as well as from a (frequency-independent) second-order generalization (see the main text for details) in comparison with numerical renormalization group data. Both for the NRG as well as for the first order FRG scheme, this quantity is constant over orders of magnitude as the charge susceptibility $\chi$ is governed by a power law of the bare hopping amplitude $t^{\prime }$.