[Paper Review] Computing local multipoint correlators using the numerical renormalization group

Seung‐Sup B. Lee, Fabian B. Kugler+1 more|arXiv (Cornell University)|Jan 3, 2021

Physics of Superconductivity and Magnetism106 references33 citations

TL;DR

This paper presents a numerical renormalization group (NRG) method to compute local three- and four-point correlators in quantum impurity models with high accuracy across all frequency and temperature regimes. By introducing partial spectral functions (PSFs) and recursive convolution with formalism-specific kernels, the approach enables nonperturbative, real-frequency computation of connected correlators and RIXS spectra, including effects from Kondo physics and Anderson orthogonality.

ABSTRACT

Local three- and four-point correlators yield important insight into strongly correlated systems and have many applications. However, the nonperturbative, accurate computation of multipoint correlators is challenging, particularly in the real-frequency domain for systems at low temperatures. In the accompanying paper, we introduce generalized spectral representations for multipoint correlators. Here, we develop a numerical renormalization group (NRG) approach, capable of efficiently evaluating these spectral representations, to compute local three- and four-point correlators of quantum impurity models. The key objects in our scheme are partial spectral functions, encoding the system's dynamical information. Their computation via NRG allows us to simultaneously resolve various multiparticle excitations down to the lowest energies. 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. We present exemplary results for the connected four-point correlators of the Anderson impurity model, and for resonant inelastic x-ray scattering (RIXS) spectra of related impurity models. Our method can treat temperatures and frequencies -- imaginary or real -- of all magnitudes, from large to arbitrarily small ones.

Motivation & Objective

• To develop a nonperturbative, numerically exact method for computing local multipoint correlators in strongly correlated quantum impurity models.
• To overcome the challenge of computing real-frequency, low-temperature 3- and 4-point functions, which are difficult for standard solvers like QMC and ED.
• To generalize the full-density-matrix NRG approach to handle ℓ-point correlators by introducing recursive computation of partial spectral functions (PSFs).
• To enable accurate computation of connected correlators and vertices across Matsubara, zero-temperature real-frequency, and finite-temperature Keldysh formalisms.
• To apply the method to compute resonant inelastic x-ray scattering (RIXS) spectra, capturing strong correlation effects such as the Kondo effect and Anderson orthogonality.

Proposed method

• The method introduces partial spectral functions (PSFs) as system-specific, generalized Lehmann representations encoding many-body spectral information.
• It extends the full-density-matrix (fdm) NRG framework to recursively compute PSFs for 3- and 4-point functions by leveraging lower-point PSFs as building blocks.
• An iterative scheme is developed to resolve regimes with widely differing frequency scales, |ωi| ≪ |ωj|, ensuring fine resolution across all energy scales.
• The PSFs are convolved with formalism-specific kernels—derived for Matsubara, zero-temperature real-frequency, and Keldysh formalisms—to obtain full correlators.
• For real-frequency correlators, PSFs are broadened numerically to account for discretization effects in the NRG flow.
• Connected parts of 4-point correlators are extracted via subtraction of disconnected contributions, with strategies to improve numerical accuracy.

Experimental results

Research questions

• RQ1 How can local three- and four-point correlators be computed accurately in the real-frequency domain at arbitrarily low temperatures?
• RQ2 Can the NRG method be generalized to compute multipoint correlators beyond two-point functions with exponential resolution at low energies?
• RQ3 How do strong correlation effects such as the Kondo effect and Anderson orthogonality manifest in resonant inelastic x-ray scattering (RIXS) spectra?
• RQ4 What is the role of partial spectral functions (PSFs) in enabling a unified, formalism-independent framework for multipoint correlators?
• RQ5 To what extent can the method reproduce analytical, perturbative, and QMC benchmarks across different formalisms and interaction regimes?

Key findings

• The method successfully computes connected four-point correlators of the Anderson impurity model, showing excellent agreement with perturbative results in the weak-coupling regime.
• For the Kondo regime, the method accurately captures the power-law behavior of the zero-temperature 4-point vertex, consistent with analytical predictions from x-ray absorption theory.
• The approach reproduces QMC results for Matsubara-frequency 4-point vertices of the Anderson impurity model at intermediate temperatures.
• In the infinite-U limit, the method reproduces the exact Keldysh 4-point vertex of the Hubbard atom, validating its accuracy in the strong-correlation limit.
• RIXS spectra computed for the Mahan impurity model and augmented Anderson model reveal clear signatures of Anderson orthogonality and Kondo screening, respectively.
• The method achieves temperature and frequency resolution far below the Kondo scale, enabling access to the low-energy dynamics of strongly correlated systems.