Adaptive logarithmic discretization for numerical renormalization group methods

TL;DR

This paper introduces an adaptive logarithmic discretization method for numerical renormalization group calculations, improving sampling accuracy near critical points like the Fermi level by dynamically adjusting discretization mesh points.

Contribution

It proposes an adaptive discretization scheme that enhances the sampling of functions with variable density, reducing numerical artifacts in NRG methods compared to traditional fixed schemes.

Findings

Reduces systematic deviations in discretization.

Improves accuracy near low-density regions.

Provides a reference implementation for the adaptive scheme.

Abstract

The problem of the logarithmic discretization of an arbitrary positive function (such as the density of states) is studied in general terms. Logarithmic discretization has arbitrary high resolution around some chosen point (such as Fermi level) and it finds application, for example, in the numerical renormalization group (NRG) approach to quantum impurity problems (Kondo model), where the continuum of the conduction band states needs to be reduced to a finite number of levels with good sampling near the Fermi level. The discretization schemes under discussion are required to reproduce the original function after averaging over different interleaved discretization meshes, thus systematic deviations which appear in the conventional logarithmic discretization are eliminated. An improved scheme is proposed in which the discretization-mesh points themselves are determined in an adaptive way; they are denser in the regions where the function has higher values. Such schemes help in reducing the residual numeric artefacts in NRG calculations in situations where the density of states approaches zero over extended intervals. A reference implementation of the solver for the differential equations which determine the full set of discretization coefficients is also described.