Hypergeometric resummation of self-consistent sunset diagrams for steady-state electron-boson quantum many-body systems out of equilibrium

A newly developed hypergeometric resummation technique [H. Mera et al., Phys. Rev. Lett. 115, 143001 (2015)] provides an easy-to-use recipe to obtain conserving approximations within the self-consistent nonequilibrium many-body perturbation theory. We demonstrate the usefulness of this technique by calculating the phonon-limited electronic current in a model of a single-molecule junction within the self-consistent Born approximation for the electron-phonon interacting system, where the perturbation expansion for the nonequilibrium Green's function in powers of the free bosonic propagator typically consists of a series of noncrossing sunset diagrams. Hypergeometric resummation preserves conservation laws and it is shown to provide substantial convergence acceleration relative to more standard approaches to self-consistency. This result strongly suggests that the convergence of the self-consistent sunset series is limited by a branch-cut singularity, which is accurately described by Gauss hypergeometric functions. Our results showcase an alternative approach to conservation laws and self-consistency where expectation values obtained from conserving divergent perturbation expansions are summed to their self-consistent value by analytic continuation functions able to mimic the convergence-limiting singularity structure.

Figure 1

Diagrammatic representation of the interacting NEGF (double straight line) within the Fock-SCBA. This NEGF is asymptotic to a series in powers of the noninteracting NEGF (single straight line) and the boson propagator (semicircle). The brackets enclose all sunset diagrams contributing to each order in the expansion in powers of $U^2$ (square of the strength of electron-boson interaction).

Figure 2

Schematic view of the single-molecule junction model driven by finite bias voltage ${\mu }_L-{\mu }_R=V_b$. Here ${\gamma }_{L/R}$ is the tunneling rate in and out of the molecular electronic energy levels; $\mathrm{\Delta }$ is the energy gap between neighboring levels; $\mathrm{\Omega }$ is the phonon frequency of the single vibrational mode taken into account; and $U$ is the strength of electron-phonon interaction. (b) Current-voltage characteristics of the single-molecule junction in (a) in the absence $(U=0$, dashed line) and presence $(U=4.0{\gamma }_R$, filled dots) of electron-phonon interaction. The latter case is computed using Fock-SCBA. The current is normalized by dividing by its noninteracting value at $V_b=7.5\mathrm{\Delta }$. For the model considered the interaction between electrons and phonon reduces the magnitude of the current relative to the noninteracting case.

Figure 3

(a) Phonon-induced degradation of the electronic current as a function of the electron-phonon interaction strength $U$, calculated within the standard Fock-SCBA (dots) or perturbative Fock-SCBA (thin lines). Perturbation theory reproduces the self-consistent result only for very weak interactions as it diverges for $U>U_c\approx {\gamma }_L$. The perturbatively calculated current degradation is unphysical because it oscillates wildly as higher-order corrections are added to the series. In contrast, the standard Fock-SCBA approach based on Eq. (4) yields a regularized converged result. (b) Number of iterations of Eq. (4) needed to converge the Fock-SCBA current. The vertical dotted black lines in both panels indicate the apparent radius of convergence, $U_c\approx {\gamma }_L$, of the perturbation series for electronic current. The bias voltage is set as $V_b=7.5\mathrm{\Delta }$.

Figure 4

Phonon-induced current degradation as a function of $U$ calculated by the standard SCBA iteration given by: Eq. (4) (dots); the hypergeometric approximant given by Eq. (9) (thick solid line); the hypergeometric approximant given by Eq. (17) (thick dotted line); the 1/1 Padé approximant (thin dotted line); and the 2/2 Padé approximant (thin dot-dashed line). For the range of values of $U$ considered both hypergeometric approximants give excellent approximations to the self-consistent result, as well as a substantial improvement over Padé approximants. The bias voltage is set as $V_b=7.5\mathrm{\Delta }$.

Figure 5

I-V characteristics calculated by hypergeometric resummation (thick solid and dashed lines) is compared with standard Fock-SCBA (dots). For reference, we give also the noninteracting current (thin dashed line). The current has been normalized to its noninteracting value at $V_b=7.5\mathrm{\Delta }$. The value of the interaction strength is $U/{\gamma }_R=4$. The hypergeometric estimates were obtained in just 54 iterations of Eq. (6)—a dramatic reduction relative to standard Eq. (4).

Figure 6

Imaginary part of the phonon-limited electronic current at high bias voltage $V_b=7.5\mathrm{\Delta }$ in the model of single-molecule junction from Fig. 2 as a function of complex $U$. The sign of the imaginary part of the current on each quadrant is indicated by $\pm$. The physical current is found for $\mathrm{Im}U=0$ where the imaginary part of the current is zero. The current is calculated by summing the self-consistent sunset series in Fig. 1 by means of: (a) the hypergeometric approximant given by Eq. (9); (b) the hypergeometric approximant given by Eq. (17); (c) the 1/1 Padé approximant; (d) the 2/2 Padé approximant. In (a) and (b) the convergence-limiting singularity is a branch cut along $\mathrm{Re}U=0$; the imaginary part of the current discontinuously changes sign across the imaginary axis. In (c) and (d) the convergence limiting singularity is the closest pole to the origin. Padé approximants attempt to reproduce the cut by a line of poles—the 1/1 Padé approximant has two poles, while the 2/2 Padé approximant has four.