Nonequilibrium dual-boson approach

We develop a nonequilibrium auxiliary quantum master equation dual-boson method (aux-DB) and argue that it presents a convenient way to describe steady states of correlated impurity models (such as single-molecule optoelectronic devices) where electron and energy transport should be taken into account. The aux-DB is shown to provide high accuracy with relatively low numerical cost. Theoretical analysis is followed by illustrative simulations within generic junction models, where the scheme is benchmarked against numerically exact results.

Figure 1

Nonequilibrium junction model. Shown are (a) physical model and (b) reference system within aux-DB approach.

Figure 2

Electron $I_L$ and phonon $J_P$ fluxes. Shown are results for the (a) RLM and (b) AIM. In (a), aux-DB results (solid blue line) are benchmarked vs the numerically exact QMC calculation of Ref. [53]. (b) Comparison of the aux-DB results for the AIM with $U=0$ and $U=7.5$. The solid blue line presents the RLM simulations within aux-DB and is the same in both panels.

Figure 3

Spectral function $A$ for the AIM with (a) symmetric ($U=0, V_{sd}=6$) and (b) antisymmetric ($U=5, V_{sd}=0$) couplings to the thermal bath $P$. Shown are results of the auxiliary QME (dotted red line), zero-order (dashed green line), and first-order (solid blue line) aux-DB approaches. The inset in (a) shows the aux-DB results for the RLM.

Figure 4

Spectral function $A$ of the AIM with antisymmetric coupling to thermal bath $P$. Shown are results for several molecule-thermal bath coupling strengths, with (a) destruction of the Kondo peak by dephasing induced by coupling to thermal bath and (b) broadening of the Coulomb peaks (for comparison, all Coulomb peaks are shifted to a common maximum set at $E=0$).

Figure 5

Contributions to diagrams for dual fermion, ${\mathrm{\Sigma }}^{DF}$, and dual boson, ${\mathrm{\Pi }}^{DB}$, self-energies, given by Eq. (6). Directed solid and wavy lines (black) indicate dual-fermion and dual-boson GFs, $G_0^{DF}$ and $D_0^{DB}$, respectively. The triangles and square (blue) indicate vertices $\delta , \gamma$ and $\mathrm{\Gamma }$ of the reference system.

Figure 6

Hybridization functions of the physical (solid blue line) and auxiliary (dashed red line) systems. Shown are the (a) retarded and (b) Keldysh projections of the self-energy due to coupling to contacts and (c) hybridization function due to coupling to the thermal bath. Fitting is done for parameters adopted in the first numerical example presented in the main text.