[Paper Review] Improved estimator for numerical renormalization group calculations of the self-energy

Fabian B. Kugler|arXiv (Cornell University)|Feb 8, 2022

Quantum and electron transport phenomena46 references22 citations

TL;DR

This paper introduces a new self-energy estimator for numerical renormalization group (NRG) calculations that combines two equations of motion to eliminate artifacts in the imaginary part of the retarded self-energy. The method ensures exact high-frequency normalization, suppresses unphysical overshoots and wiggles at low energies, and achieves converged results with fewer kept states, significantly improving accuracy and efficiency for multiorbital DMFT calculations.

ABSTRACT

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. 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 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. 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. 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.

Motivation & Objective

• Address persistent artifacts in NRG self-energy calculations, such as unphysical overshoots and wiggles in the imaginary part of the retarded self-energy.
• Resolve normalization issues in the high-frequency asymptote of the real part of the self-energy, which can be violated in standard NRG implementations.
• Reduce the numerical cost of accurate self-energy calculations by enabling convergence with fewer kept states in the NRG calculation.
• Provide a robust, generalizable framework for self-energy estimation in quantum impurity models, especially within DMFT and multiorbital systems.
• Improve the reliability and efficiency of NRG as an impurity solver for strongly correlated materials and lattice models.

Proposed method

• Propose a new self-energy estimator based on a combined application of one- and two-fold equations of motion, avoiding reliance on Kramers–Kronig transforms of real parts.
• Derive a formula involving three correlators (G, F, I) that directly determines the imaginary part of the self-energy from auxiliary correlators.
• Ensure exact fulfillment of the high-frequency asymptote of ReΣ by construction, using the spectral weight conservation of fdm NRG.
• Use the imaginary parts of auxiliary correlators F and I directly, bypassing the need for real parts obtained via Kramers–Kronig, which are prone to numerical inaccuracies.
• Implement the new estimator in a way that maintains symmetry and numerical stability, especially in matrix-valued cases.
• Validate the method across multiple models: single-orbital Anderson, and one-, two-, and three-orbital Hubbard models in DMFT.

Experimental results

Research questions

• RQ1 Can a new self-energy estimator be derived that avoids the unphysical overshoots and wiggles in ImΣν observed in standard NRG calculations?
• RQ2 Does the new estimator ensure exact normalization of the high-frequency asymptote of ReΣ, even in challenging multiorbital regimes?
• RQ3 Can the new method achieve converged results for the self-energy with fewer kept states compared to the standard estimator?
• RQ4 How does the new estimator perform in particle-hole symmetric and strongly correlated regimes, particularly at low energies?
• RQ5 To what extent can the new estimator be generalized to matrix-valued correlation functions in multiorbital systems?

Key findings

• The new estimator, ΣIFG, eliminates unphysical overshoots in ImΣν at low energies, which were previously observed in the standard NRG estimator ΣFG.
• The imaginary part of the self-energy, −ImΣIFGν, exhibits a clean, non-negative parabolic shape with its vertex exactly at the origin, consistent with causality and Fermi-liquid theory.
• For the same level of accuracy, the new estimator converges with significantly fewer kept states—e.g., Nkp = 500 suffices for converged ImΣIFGν, while Nkp = 2000 was needed for ΣFG to approach convergence.
• The high-frequency asymptote of ReΣ is exactly preserved by the new estimator, unlike the standard method, which can violate this analytical constraint.
• The value of ImΣIFG(0) is improved by several orders of magnitude compared to ΣFG, approaching the exact value of zero in particle-hole symmetric cases.
• The method reduces numerical effort by enabling accurate results at lower Nkp, making it highly efficient for multiorbital DMFT applications.