Anderson Orthogonality and the Numerical Renormalization Group

TL;DR

This paper demonstrates how the numerical renormalization group (NRG) can accurately compute Anderson Orthogonality exponents by analyzing the exponential decay of ground state overlaps, validated across various interacting models.

Contribution

It introduces a simple NRG-based method to precisely calculate Anderson Orthogonality exponents from ground state overlaps, aligning with phase shift and charge displacement measures.

Findings

NRG overlap decay is exponential with chain length, enabling exponent extraction.

Calculated exponents agree within 1% with phase shift and charge displacement methods.

Method applies to complex interacting models, including those with population switching.

Abstract

Anderson Orthogonality (AO) refers to the fact that the ground states of two Fermi seas that experience different local scattering potentials, say |G_I> and |G_F>, become orthogonal in the thermodynamic limit of large particle number N, in that |<G_I|G_F>| ~ N^(- Delta_AO^2 /2) for N->infinity. We show that the numerical renormalization group (NRG) offers a simple and precise way to calculate the exponent Delta_AO: the overlap, calculated as function of Wilson chain length k, decays exponentially, ~ exp(-k alpha), and Delta_AO can be extracted directly from the exponent alpha. The results for Delta_AO so obtained are consistent (with relative errors typically smaller than 1%) with two other related quantities that compare how ground state properties change upon switching from |G_I> to |G_F>: the difference in scattering phase shifts at the Fermi energy, and the displaced charge flowing in from infinity. We illustrate this for several nontrivial interacting models, including systems that exhibit population switching.