Fluctuation-exchange approximation theory of the nonequilibrium singlet-triplet transition

As a continuation of a previous work [B. Horváth et al., Phys. Rev. B 82, 165129 (2010)], here we extend the so-called fluctuation exchange approximation (FLEX) to study the nonequilibrium singlet-triplet transition. We show that, while being relatively fast and a conserving approximation, FLEX is able to recover all important features of the transition, including the evolution of the linear conductance throughout the transition, the two-stage Kondo effect on the triplet side, and the gradual opening of the singlet-triplet gap on the triplet side of the transition. A comparison with numerical renormalization-group calculations also shows that FLEX captures rather well the width of the Kondo resonance. FLEX thus offers a viable route to describe correlated multilevel systems under nonequilibrium conditions, and in its rather general form, as formulated here, it could find a broad application in molecular electronics calculations.

Figure 1

The $\Phi$ functional generating the FLEX diagrams. The first diagram just generates the Hartree-Fock approximation. Thick lines denote full Green's functions. Squares denote the particle-hole vertex, defined in Eq. (20).

Figure 2

Structure of the particle-hole vertex, ${\tilde{\Gamma }}_{\alpha \gamma }^{\beta \delta }$. Here $\alpha$, $\beta$, $\gamma$, and $\delta$ denote composite indices as introduced in Eq. (21).

Figure 3

(a) Series of self-energy diagrams generated from ${\Phi }_{\mathrm{FLEX}}$. (b) Diagrammatic definition of the full particle-hole propagator, $\Pi$.

Figure 4

Total spectral function, ${\rho }_T\left(\omega \right)=({\rho }_{+}\left(\omega \right)+{\rho }_{-}\left(\omega \right))/2$ for $J/U=0$ and ${\Gamma }_{\pm }/U=0.785$ for different values of level splitting, $\Delta /U$. Top: FLEX results. Bottom: IPT results for the same parameters.

Figure 5

Total equilibrium spectral function ${\rho }_T\left(\omega \right)$ for $J/U=0.15$ and ${\Gamma }_{\pm }/U=0.785$, for different values of level splitting, $\Delta /U$, as computed by FLEX (top) and by IPT (bottom).

Figure 6

Comparison of the spectral functions for $\Delta =0$, ${\Gamma }_{\pm }/U=0.785$, and $J/U=0.15$, as computed by FLEX, IPT, and NRG. Clearly, FLEX seems to capture rather accurately the width of the central Kondo resonance.

Figure 7

Level-resolved (top) and total (bottom) equilibrium spectral functions for $J/U=0.15$, ${\Gamma }_{+}/U=1.1$, and ${\Gamma }_{-}/U=0.785$ for different level splittings, $\Delta /U$, as computed by FLEX. Inset (top): Normalized spectral functions ${\rho }_{\pm }\left(\omega \right)/{\rho }_{\pm }\left(0\right)$, demonstrating the presence of the two different Kondo scales.

Figure 8

Differential conductance, $G\left(V\right)=\mathrm{d}I/\mathrm{d}V$, in units of $e^2/h$, for lateral (left panels) and vertical (right panels) dots, with $J/U=0.15$, ${\Gamma }_{+}/U=0.785$, and ${\Gamma }_{-}/U=1.1$ for different level splittings, $\Delta /U$, as obtained by FLEX. On the triplet side ($\Delta =0$), the second Kondo scale emerges as a narrow dip/sharp resonance in $G\left(V\right)$ in the lateral/vertical arrangement. For large $\Delta$'s the singlet-triplet splitting gives rise to a wide central dip in $G\left(V\right)$. The curves reproduce very nicely all features observed experimentally, however, the crossover regime is only qualitatively captured in the lateral case.

Figure 9

Linear conductance, for lateral and vertical dots, with $J/U=0.15$, ${\Gamma }_{+}/U=0.785$, and ${\Gamma }_{-}/U=1.1$ for different level splittings, $\Delta /U$, as obtained by FLEX.

Figure 10

Differential conductance (in units of $e^2/h$) of a vertical and a lateral dot with $J/U=0.15$, ${\Gamma }_{+}/U=1.1$, ${\Gamma }_{-}/U=0.785$, and $\Delta =0$, as obtained by IPT. The curves compare quite well with those in Fig. 8.

Figure 11

Equilibrium dimensionless spectral functions for $J/U=0.15$ ${\Gamma }_{+}/U=1.1$, ${\Gamma }_{-}/U=0.785$, and $\Delta /U=0$ for a symmetrical dot with ${\phi }_{\pm }=\pm \pi /4$ and for an asymmetrical dot with ${\phi }_{\pm }=\pm \pi /3$, as computed by FLEX. Inset: The of-fdiagonal component of the spectral function for ${\phi }_{\pm }=\pm \pi /3$.

Figure 12

Top: Differential conductance, $G\left(V\right)=\mathrm{d}I/\mathrm{d}V$, for lateral (top) and vertical (bottom) dots with $J/U=0.15$, ${\Gamma }_{+}/U=0.785$, ${\Gamma }_{-}/U=1.1$, and $\Delta /U=0$, as obtained by FLEX, for the symmetrical case, ${\phi }_{\pm }=\pm \pi /4$, and for the asymmetrical case, ${\phi }_{\pm }=\pm \pi /3$.