Anderson-Holstein model in two flavors of the noncrossing approximation

The dynamical interplay between electron-electron interactions and electron-phonon coupling is investigated within the Anderson-Holstein model, a minimal model for open quantum systems that embody these effects. The influence of phonons on spectral and transport properties is explored in equilibrium, for nonequilibrium steady state and for transient dynamics after a quench. Both the particle-hole symmetric and the more generic particle-hole asymmetric cases are studied. The treatment is based on two complementary noncrossing approximations, the first of which is constructed around the weak-coupling limit and the second around the polaron limit. In general, the two methods disagree in nontrivial ways, indicating that more reliable approaches to the problem are needed. The frameworks used here can form the starting point for numerically exact methods based on bold-line continuous-time quantum Monte Carlo algorithms capable of treating open systems simultaneously coupled to multiple fermionic and bosonic baths.

Figure 1

(a) The elements of the unperturbed propagator $G_{\alpha \alpha }^{\left(0\right)}$ (left column, thin lines) and of the NCA propagator $G_{\alpha \alpha }$ (right column, bold lines). The upper line represents spin up occupation and the lower line spin down occupation. The dotted line signifies that the spin level is unoccupied, while a solid line marks it as occupied. (b) An example of a diagram included in the reduced propagator $G_{00}$. Electronic hybridization lines are shown as wiggly lines and phonon interaction lines as gluon lines. (c) An example of a diagram on the Keldysh contour with interbranch lines and a special hybridization line ending at the final time, corresponding to a contribution to the current.

Figure 2

The electron hybridization diagrams included in the bare NCA self energy, where the wiggly lines denote electronic dot-lead hybridization lines. The pairs of straight lines represent the dot's electronic state, with the two lines standing for the two possible spins: A solid line represents an occupied spin level, whereas dashed lines stand for empty spin levels.

Figure 3

The phonon interaction diagrams for the bare NCA self energy in the symmetric case $\delta =1$. The curly lines denote phonon interaction lines, and straight lines are as in Fig. 2.

Figure 4

(a) The diagrams representing the different matrix elements of the dressed NCA self energy. The wiggly double lines denote electron hybridization lines dressed by phonon interactions. (b) An example of a bare NCA diagram of the lowest order is not included in the dressed NCA diagrams.

Figure 5

The phonon interaction diagrams for asymmetrical model.

Figure 6

(left panels) The time evolution of the spectral function $A(\omega ;t)$ within the bare NCA is shown for different phonon frequencies. (right panels) Time dependence of cuts at $\omega =0$ (blue) and $\omega =U/2$ (green). The time scale $2\pi /{\omega }_0$ related to the phonon frequency is also plotted for comparison. A symmetric dot with $U=-2\epsilon =10\mathrm{\Gamma }$ is considered at equilibrium $V=0$. The phonon coupling is set to $\lambda =1.5\mathrm{\Gamma }$ and the counter term is symmetric $\left(\delta =1\right)$. The inverse temperature of all baths is $\beta =10/\mathrm{\Gamma }$.

Figure 7

The same as Fig. 6 within the dressed NCA. A symmetric dot with $U=-2\epsilon =10\mathrm{\Gamma }$ is considered at equilibrium $V=0$. The phonon coupling is symmetric with $\lambda =1.5\mathrm{\Gamma }$ and the inverse temperature of all baths is $\beta =10/\mathrm{\Gamma }$.

Figure 8

The ${\omega }_0$ dependence of the spectral function $A\left(\omega \right)$ is calculated by (a) bare NCA and (b) dressed NCA for a symmetric dot at equilibrium $V=0$ with $U=-2\epsilon =10\mathrm{\Gamma }$. The phonon coupling is $\lambda =1.5\mathrm{\Gamma }$ and the counter term is symmetric $\left(\delta =1\right)$. All baths at the same inverse temperature $\beta =10/\mathrm{\Gamma }$. The dashed lines indicate the renormalized charge transfer peak at ${\omega }_{\pm }=\pm \left(\epsilon +\frac{{\lambda }^2}{{\omega }_0}\right)$. The ${\omega }_0$ dependence of the central peak at $\omega =0$ is plotted in (c).

Figure 9

The $\lambda$ dependence of the spectral function $A\left(\omega \right)$ as calculated within the (a) bare NCA and (b) dressed NCA for a symmetric dot with $U=-2\epsilon =10\mathrm{\Gamma }$ at equilibrium $V=0$. The phonon coupling is ${\omega }_0=1.0\mathrm{\Gamma }$ and the counter term is symmetric $\left(\delta =1\right)$. All baths at the same inverse temperature $\beta =10/\mathrm{\Gamma }$. The dashed lines indicate the renormalized charge transfer peak at ${\omega }_{\pm }=\pm \left(\epsilon +\frac{{\lambda }^2}{{\omega }_0}\right)$. The $\lambda$ dependence of the central peak at $\omega =0$ is plotted in (c).

Figure 10

The ${\omega }_0$ dependence of the spectral function $A\left(\omega \right)$ for a symmetric dot with $U=-2\epsilon =10\mathrm{\Gamma }$ under a nonequilibrium symmetrically applied bias voltage $V=2\mathrm{\Gamma }$ within the (a) bare NCA and (b) dressed NCA. The phonon coupling is $\lambda =1.5\mathrm{\Gamma }$ and the counter term is symmetric $\left(\delta =1\right)$. All baths at the same inverse temperature $\beta =10/\mathrm{\Gamma }$.

Figure 11

(left panels) The time evolution of the spectral function $A(\omega ;t)$ within the bare NCA is shown for different phonon frequencies. The frequency oscillations of the CT peaks along with an illustration of the expected energy oscillations in the adiabatic limit (dash lines) are also exhibited. (right panels) Time dependence of cuts at $\omega =0$ (blue) and $\omega =U/2$ (green). The time scale $2\pi /{\omega }_0$ related to the phonon frequency is also plotted for comparison. The dot is symmetric with $U=-2\epsilon =10\mathrm{\Gamma }$ at equilibrium $V=0$. The phonon coupling is $\lambda =1.5\mathrm{\Gamma }$ and the counter term is asymmetric $\left(\delta =0\right)$. The inverse temperature of all baths is $\beta =10/\mathrm{\Gamma }$.

Figure 12

The same as Fig. 11 within the dressed NCA. The dot is symmetric with $U=-2\epsilon =10\mathrm{\Gamma }$ at equilibrium $V=0$. The phonon coupling is asymmetric $\left(\delta =0\right)$ with $\lambda =1.5\mathrm{\Gamma }$ and the inverse temperature is $\beta =10/\mathrm{\Gamma }$.

Figure 13

The ${\omega }_0$ dependence of the spectral function $A\left(\omega \right)$ for a dot in equilibrium as calculated within the (a) bare NCA and (b) dressed NCA. The electron-phonon coupling is asymmetric $\left(\delta =0\right)$ and the coupling strength is $\lambda =1.5\mathrm{\Gamma }$. The dot is symmetric with $U=-2\epsilon =10\mathrm{\Gamma }$. All baths at the same inverse temperature $\beta =10/\mathrm{\Gamma }$.

Figure 14

The $\lambda$ dependence of the spectral function $A\left(\omega \right)$ as calculated within the (a) bare NCA and the (b) dressed NCA for an equilibrium symmetric dot with $U=-2\epsilon =10\mathrm{\Gamma }$. The phonon frequency is ${\omega }_0/\mathrm{\Gamma }=2.5$. The dashed lines indicate the center of the CT peaks as estimated by the energy renormalization at the antiadiabatic limit ${\omega }_{CT}^{+}/\mathrm{\Gamma }=-\epsilon +\frac{{\lambda }^2}{{\omega }_0}$ and ${\omega }_{CT}^{-}/\mathrm{\Gamma }=\epsilon +3\frac{{\lambda }^2}{{\omega }_0}$. All baths at the same inverse temperature $\beta =10/\mathrm{\Gamma }$.

Figure 15

The conductance $G\left(V\right)$ as calculated within the bare NCA for different electron-phonon coupling (a)$\lambda /\mathrm{\Gamma }=1$ and (b)$\lambda /\mathrm{\Gamma }=2$ with a symmetrically applied bias ${\mu }_L={\mu }_R=V$. The dot is also symmetric with $U=-2\epsilon =10\mathrm{\Gamma }$. Panel (c) shows the ${\omega }_0$ dependence of the central peak at $\omega =0$. All baths at the same inverse temperature $\beta =10/\mathrm{\Gamma }$.

Figure 16

Left: Evolution of the spectral function across the metal-insulator transition (gap closing) by increasing the phonon coupling. Right: The spectral function $A\left(\omega \right)$ in the strong coupling regime is calculated within the dressed NCA for a symmetric dot with $U=-2\epsilon =10\mathrm{\Gamma }$ at equilibrium $V=0$. The density of state is of the semicircular form $\mathrm{\Gamma }\left(\omega \right)=\sqrt{4t^2-{\omega }^2}$ with $t=1$. The phonon coupling is ${\omega }_0=3.0\mathrm{\Gamma }$ and the counter term is symmetric $\left(\delta =1\right)$. The baths are maintained at a temperature $\beta \mathrm{\Gamma }=50$.