Dynamical screening effects in correlated materials: Plasmon satellites and spectral weight transfers from a Green's function ansatz to extended dynamical mean field theory

Dynamical screening of the Coulomb interactions in correlated electron systems results in a low-energy effective problem with a dynamical Hubbard interaction $\mathcal{U}\left(\omega \right)$. We propose a Green's function ansatz for the Anderson impurity problem with retarded interactions, in which the Green's function factorizes into a contribution stemming from an effective static-$U$ problem and a bosonic high-energy part introducing collective plasmon excitations. Our approach relies on the scale separation of the low-energy properties, related to the instantaneous static $U$, from the intermediate- to high-energy features originating from the retarded part of the interaction. We argue that for correlated materials where retarded interactions arise from downfolding higher-energy degrees of freedom, the characteristic frequencies are typically in the antiadiabatic regime. In this case, accurate approximations to the bosonic factor are relatively easy to construct, with the most simple being the boson factor of the dynamical atomic-limit problem. We benchmark the quality of our method against numerically exact continuous-time quantum Monte Carlo results for the Anderson-Holstein model both at half- and quarter-filling. Furthermore, we study the Mott transition within the Hubbard-Holstein model within extended dynamical mean field theory. Finally, we apply our technique to a realistic three-band Hamiltonian for SrVO${}_3$. We show that our approach reproduces both the effective mass renormalization and the position of the lower Hubbard band by means of a dynamically screened $U$, previously determined ab initio within the constrained random phase approximation. Our approach could also be used within schemes beyond dynamical mean field theory, opening a quite general way of describing satellites and plasmon excitations in correlated materials.

Figure 1

Dynamic $U$ for a single plasmon mode with ${\omega }_0=15$, $U_0=3.6$, and $U_{\infty }=15$. This is the single plasmon modelization of the dynamic $U$ for SrVO${}_3$, determined ab initio by constrained RPA (Ref. 5). The function $U\left(z\right)$ is plotted on both the real ($z=\omega$) and imaginary ($z=i\omega$) frequency domains. The vertical asymptote is shown at $\omega ={\omega }_0$ for $U\left(\omega \right)$, where the function has a single pole. In the most general case, there will be as many poles as plasmon modes.

Figure 2

Green's functions for a half-filled Anderson model with dynamically screened $U$ and a semicircular density of states at $\beta =10$. The $G_{\mathrm{DALA}}$ obtained by the method proposed in Sec. 5 is plotted (green long-dashed line) and compared to the exact Rubtsov's CTQMC numerical result (blue dotted line). Also, the dynamic atomic limit (red solid line) and the static one (pink dotted-dashed line) are reported. The system is computed at a quite large temperature such that an accurate benchmark against the numerically exact CTQMC is still possible, even for quite large values of $U$.

Figure 3

Spectral function of the $G_{\mathrm{DALA}}$ Green's functions reported in Fig. 2.

Figure 4

Half-filled Anderson impurity model with retarded screened interaction: ${\omega }_0=5$, $U_{\infty }-U_0=5$, $\beta =10$, for various values of $U_0$. (a) Numerically exact Rubtsov's CTQMC and DALA $G\left(\tau \right)$; (b) CTQMC and DALA imaginary part of $G\left(i{\omega }_n\right)$.

Figure 5

Symmetric Anderson model with retarded screened interaction: $U_0=1.25,{\omega }_0=2,U_{\infty }=2.5$, at different $\beta$. (a) Imaginary part of the numerically exact Rubtsov's CTQMC and DALA Green's functions for small Matsubara frequencies ${\omega }_n$; (b) $-{\omega }_n\mathrm{Im}\left[G\left(i{\omega }_n\right)\right]$ in the intermediate frequency range. The arrow indicates the frequency when the DALA and the CTQMC Green's functions become practically indistinguishable. The DALA reproduces correctly the intermediate- to high-energy behavior of the Green's function well beyond the “trivial” $1/i\omega$ term; (c) $G\left(\tau \right)$; (d) the CTQMC and DALA Bose factor $F\left(\tau \right)$. The thicker lines are for the DALA. The discrepancy between the DALA and the exact factors is getting smaller as the temperature increases. The wiggles are due to the stochastic noise of the data.

Figure 6

Symmetric Anderson model with retarded screened interaction as in Fig. 5: $U_0=1.25,{\omega }_0=2,U_{\infty }=2.5$. (a) Imaginary part of the numerically exact Rubtsov's CTQMC and DALA $+$ LF Green's functions for small Matsubara frequencies ${\omega }_n$ at different $\beta$. (b) The CTQMC, DALA, and DALA $+$ LF Bose factor $F\left(\tau \right)$ at $\beta =160$. The improvement of the $F\left(\tau \right)$ provided by the Lang-Firsov correction is apparent. The wiggles are due to the interpolation of noisy QMC data. (c) Spectral representation of the DALA $+$ LF Green's function in (a).

Figure 7

Symmetric Anderson model with retarded screened interaction as in Fig. 4: ${\omega }_0=5$, $U_{\infty }-U_0=5$, $\beta =10$, for $\beta =20$ and two values of $U_0$. (a) $U_0=1$; (b) $U_0=2$. The numerically exact Rubtsov's CTQMC, DALA, DALA $+$ LF, GC, and IBC Green's functions are reported.

Figure 8

Asymmetric spin unpolarized Anderson model with retarded screened interaction as in Fig. 7: ${\omega }_0=5$, $U_{\infty }-U_0=5$, for $\beta =20$ and two values of $U_0$. (a) $U_0=1$; (b) $U_0=2$. In this case, the number of particles $N_{\sigma }$ was set to be around the quarter-filling. The numerically exact Rubtsov's CTQMC, DALA, DALA $+$ LF, GC, and IBC Green's functions are reported.

Figure 9

Critical $U_{\mathrm{c}2}$ line of the Mott transition calculated at $\beta =40$ and half-filling on a Bethe lattice for the screened instantaneous interaction $U_0$ as a function of $U_{\infty }/U_0$ and ${\omega }_0=10,20$ (green diamonds and red circles, respectively). The critical values are obtained from the Green's function computed by the DALA $+$ LF ansatz. The behavior of the critical screened $U$ is in agreement with the one computed by Werner and Millis with their CTQMC algorithm at ${\omega }_0=10$ and $\beta =100$ (black upper triangles). Indeed, the critical line calculated by means of our DALA $+$ LF approach (blue lower triangles) is on top of their CTQMC points (Ref. 21). This highlights the accuracy of our approximated DALA $+$ LF method if compared to the exact numerical result. Note the quite strong temperature dependence of the critical $U_{\mathrm{c}2}$ in going from $\beta =40$ to 100. This is a quite interesting effect, which deserves further analysis.

Figure 10

Spectral representation of the Green's function for a Holstein-Hubbard model with ${\omega }_0=10$, $U_0=2$, $U_{\infty }=6.5$, and $\beta =100$, close to the blue Mott transition line of Fig. 9. The analytic continuation has been assisted by the DALA Bose factor, which allows one to describe accurately the plasmon satellites centered around frequencies multiple of ${\omega }_0$.

Figure 11

Spectral function of one electron in 3 $t_{2g}$ bands for the SrVO${}_3$ obtained by the DFT in the local density approximation. The DMFT calculations are done at $\beta =10$ eV${}^{-1}$ with the DALA Bose factor solver for the retarded $U$ and the Hirsch-Fye QMC solver for the instantaneous interaction. Three types of interactions are reported: the purely static one, with $U_0=3.6$ eV, a dynamic Holstein one with the same instantaneous interaction but with the unscreened $U_{\infty }=7$ eV, and a plasmon frequency ${\omega }_0=5$ eV, another Holstein interaction with the same $U_0$ and $U_{\infty }$ but different ${\omega }_0(=15\mathrm{eV})$. Note the displacement of the upper Hubbard shoulder shifted at lower energy by the presence of the plasmon at ${\omega }_0=5$ eV.

Figure 12

Spectral function at $\beta =10$ eV${}^{-1}$ of one electron in 3 $t_{2g}$ bands for the SrVO${}_3$ obtained by the DFT in the local density approximation. The DMFT calculations are carried out with the DALA Bose factor solver for the retarded $U$ and the Hirsch-Fye QMC solver for the instantaneous interaction. The result of the realistic Green's function is reported, showing a more correlated behavior than the corresponding static model with the same instantaneous $U_0(=3.6\mathrm{eV})$. The quasiparticle peak is smaller than the corresponding static model, with a spectral weight transfer at higher energies.