By subsequently convolving the partial spectral functions with appropriate kernels, we obtain multipoint correlators in the imaginary-frequency Matsubara, the real-frequency zero-temperature, and the real-frequency Keldysh formalisms.
![qspace nrg qspace nrg](https://static3.seekingalpha.com/uploads/sa_presentations/544/62544/slides/28.jpg)
Their computation via numerical renormalization group allows us to simultaneously resolve various multiparticle excitations down to the lowest energies. The key objects in our scheme are partial spectral functions, encoding the system’s dynamical information. Here, we develop a numerical renormalization group approach, capable of efficiently evaluating these spectral representations, to compute local three- and four-point correlators of quantum impurity models. In the accompanying paper, we introduce generalized spectral representations for multipoint correlators. However, the nonperturbative, accurate computation of multipoint correlators is challenging, particularly in the real-frequency domain for systems at low temperatures. Local three- and four-point correlators yield important insight into strongly correlated systems and have many applications. We also compare our findings with previous slave-spin studies. We present analytical arguments supported by numerical results using the numerical renormalization group as DMFT impurity solver. Accordingly, it is part of a coherence-incoherence crossover and not a quantum critical point. Within this framework, the OSMP with interorbital hopping may thus reach down to extremely low temperatures $T$ but not to $T=0$. We further show that the coherence scale below which all electrons are itinerant is very small and gets exponentially suppressed even if the interorbital hopping is not overly small. Under fairly general circumstances, this implies that, at zero temperature, the OSMP, involving the Mott insulating state of one orbital, is unstable against interorbital hopping to another, metallic orbital.
![qspace nrg qspace nrg](https://i.ytimg.com/vi/L5UiYYyZD74/maxresdefault.jpg)
Here, we show how nonlocal interorbital hopping leads to local hybridization in single-site dynamical mean-field theory (DMFT). Theoretically, the OSMP is widely studied for kinetically decoupled orbitals, but the effect of interorbital hopping remains unclear. Recently, there is great interest in multiorbital systems where this transition can be restricted to certain orbitals, leading to an orbital-selective Mott phase (OSMP). The localization-delocalization transition is at the heart of strong correlation physics.
![qspace nrg qspace nrg](https://i.ytimg.com/vi/QrWS_T-NrG8/maxresdefault.jpg)
Our analysis is targeted at NRG treatments of quantum impurity models, especially those arising within dynamical mean-field theory, but most results can be straightforwardly generalized to other impurity or cluster solvers. Furthermore, we find that the new estimator yields converged results with reduced numerical effort (for a lower number of kept states) and thus is highly valuable when applying NRG to multiorbital systems. We show that the new estimator resolves the artifacts in these properties as they can be determined directly from the imaginary parts of auxiliary correlators and do not involve real parts obtained by Kramers-Kronig transform.
![qspace nrg qspace nrg](https://saure.org/cq-nrw/wp-content/uploads/2020/04/gezippt-1-768x854.jpg)
In challenging regimes, NRG results from the standard estimator, a ratio of two correlators, often suffer from artifacts: the imaginary part of the retarded self-energy is not properly normalized and, at low energies, overshoots to unphysical values and displays wiggles. We present a new estimator for the self-energy based on a combination of two equations of motion and discuss its benefits for numerical renormalization group (NRG) calculations.