Static treatment of dynamic interactions in the single-orbital Anderson impurity model

TL;DR

This work benchmarks the validity of mapping a dynamic, frequency-dependent interaction $U(\omega)$ to a static $U^*$ in a controlled single-orbital Anderson impurity model. Using the mRPA approach, which fixes $U^*$ by matching equal-time correlations via the RPA susceptibility, the authors show that a static mapping can reproduce key observables, notably the double occupancy, and often outperforms the naive choice $U^*=U(i\omega_0)$, especially at large $\omega_p$ and in strongly correlated metallic regimes. However, away from half-filling and in weaker-correlation or high-dynamic-screening regimes, a single $U^*$ cannot capture all physics, with temporal screening effects leading to features like temporal phase separation that are beyond a one-parameter effective model. The results guide practical downfolding in first-principles calculations, highlighting both the utility and limitations of mRPA for incorporating dynamic screening into static low-energy models and suggesting directions for extending to multi-orbital systems and more sophisticated renormalizations.

Abstract

Correlated electron physics is intrinsically a multiscale problem, since high-energy electronic states screen the interactions between the correlated electrons close to the Fermi level, thereby reducing the magnitude of the interaction strength and dramatically shortening its range. Thus, the handling of screening is an essential ingredient in the first-principles modelling of correlated electron systems. Screening is an intrinsically dynamic process and the corresponding downfolding methods such as the constrained Random Phase Approximation indeed produce a dynamic interaction. However, many low-energy methods require an instantaneous interaction as input, which makes it necessary to map the fully dynamic interaction to an effective instantaneous interaction strength. It is a priori not clear if and when such an effective model can capture the physics of the one with dynamic interaction and how to best perform the mapping. Here, we provide a systematic benchmark relevant to correlated materials, in the form of the Anderson impurity model. Overall, we find that a static approximation can be valid and that the moment-based approach recently proposed by Scott and Booth can be a good tool to find the value of the static interaction. We also identify physical regimes, especially under doping, where an instantaneous interaction cannot capture all of the relevant physics.

Figures (14)

• Figure 1: Left: Single-impurity Anderson model with dynamic interaction $U(\omega)$. Right: Dynamic interaction in the Matsubara representation. The three colored lines correspond to interactions with the same $U_\infty$ and $\alpha$ but different $\omega_p$ (indicated by small vertical arrows). Due to this parametrization, all three curves have the same values at $n=0$ and $n\rightarrow \infty$.
• Figure 2: Observables in the half-filled system at $\beta=10$. (a) Color maps in the $U_\infty-\alpha$ plane, the black diagonal line shows $U_\infty-\alpha=U(i\omega_0)=0$ and the dotted lines are parallel to this line. The color scale for the double occupancy is restricted to show more detail in the region of interest. (b) Observables as a function of mRPA moment, Eq. \ref{['eq:mrpa:moment']}, at $\alpha=3$. Note that a larger $U_\infty$ corresponds to a smaller mRPA moment, so the correlation strength increases going to the left in this plot. $\alpha=0$ reference results are shown in black. The mRPA works whenever the colored and black lines are on top of each other.
• Figure 3: Observables in the half-filled system at $\beta=2$ (a) and $\beta=50$ (b). The color scales are the same as in Fig. \ref{['fig:halffilled:colormap:beta10']}.
• Figure 4: Observables in the doped system at electron density $n\approx 0.8$, $\beta=10$. (a) color map (b) mRPA mapping at $\alpha=3$.
• Figure 5: Green's function $G(\tau)$ and density-density correlation function $\chi(\tau)$ shown on long (left) and short (right) imaginary time scales, in the doped system ($n\approx 0.8$) for $\omega_p=0.2$ and $\beta=10$.