Motivation and Question: Being able to predict electronic excitations is essential for making quantum technologies a reality, in particular in the context of quantum information storage and manipulation. However, it's a significant challenge to describe the strong quantum interactions in these materials accurately because their properties are governed largely by electron-electron interactions. Further, these systems often harbor localized states in which electrons are strongly correlated. Here, the typical energy scale of the electron-electron interactions becomes comparable to or larger than that of the kinetic energy of the electrons. Widely used computational methods like density functional theory (DFT) and many-body perturbation theory fail to predict the electronic behavior of these materials; developing new methodologies is thus of utmost importance.

How can we develop novel ways to theoretically describe the interactions in these complex materials without loss of accuracy and at a low computational cost? We combine the state-of-the-art many-body diagrammatic methods with new downfolding techniques that describe the excitations in terms of the excited state dynamics, corresponding to the propagation of “quasiparticles” through the system. Our methods are in quantitative agreement with experiments and can directly provide access to the spectroscopic properties of materials.

Dynamical Downfolding Methodology: When is it meaningful to distinguish different parts of a system? Roughly speaking, one or more portions must exhibit aspects that are not present or not important in the rest of the system. Fortunately, this means the strongly coupled electrons (which are key to the properties of interest) occupy, in many cases, only a small number of electronic states, which are, however, still coupled to the environment, i.e., the remaining (weakly interacting) electrons. We imagine the excitations of these correlated electrons to no longer behave like free particles but be affected (or "renormalized") by their environment. Moreover, the challenge lies in accurately capturing the "dynamical," i.e., time-dependent, renormalization of all one and two-electron interactions by the host material without any empirical parameters.

Inspired by embedding theories, wherein a small explicitly correlated region is embedded in the remaining portion of the system, and the interactions are treated at a more approximate level, we adopt a different approach. Here, we combine stochastic many-body approaches and downfold the environment, resulting in an effective dynamical many-body Hamiltonian with highly reduced dimensionality. Here, we take the dynamical quantum fluctuations from the weakly interacting environment fully into account. Specifically, we map the correlated subspace (associated with the defect) onto a general effective Hamiltonian, we determine its individual terms from one and two-body quasiparticle evolution with renormalization driven by the environment, and solve this reduced description by exact diagonalization. The dimensionality reduction of the full many-body problem is compensated by the introduction of dynamical terms, which can be viewed as a lossless compression of the full Hamiltonian.

Results and Outlook:

We exemplify our methodology on the NV- center in diamond where the defect consists of a small number of correlated states, formed by dangling bonds pointing toward the vacancy (shown in Fig. 1a) and we study the optical excitation energies obtained by exact diagonalization (shown in Fig. 1b). We can clearly see that our methodology, for both the triplet-triplet vertical transition, ³E↔³A₂ and the singlet-singlet ¹A₁↔¹E excitation, yields results that are in excellent agreement with experiment. We find that for the single-singlet transition, the screening significantly decreases the magnitude of W and enhances the singlet excitation energy, which was in previous calculations underestimated by up to 50%.

The results for the NV- center in bulk diamond practically demonstrate that the dynamically downfolded representation provide an excellent (and unprecedented) agreement with the experimentally measured excited energies of the defect. We believe that this is a jumping-off point for future practical simulations of electronic excitations in localized quantum states. The approach is general and is thus applicable to complex materials exhibiting strongly localized correlated states, e.g., in twisted bilayers and or topological insulators.