Exactly solvable toy models of unconventional magnetic alloys: Bethe Ansatz versus Renormalization Group method

TL;DR

This paper compares Bethe Ansatz and Renormalization Group methods in exactly solvable toy models of unconventional magnetic alloys with energy-dependent density of states and hybridization.

Contribution

It introduces toy models with energy-dependent parameters and demonstrates the importance of inverse band dispersion in exact solutions, contrasting with RG assumptions.

Findings

Bethe Ansatz solution highlights the role of inverse band dispersion.

RG approach assumes physics governed solely by energy-independent coupling.

Exact solutions reveal limitations of RG assumptions in these models.

Abstract

We propose toy models of unconventional magnetic alloys, in which the density of band states, $\rho(\epsilon)$, and hybridization, $t(\epsilon)$, are energy dependent; it is assumed, however, that $t^2(\epsilon)\propto\rho^{-1}(\epsilon)$, and hence an effective electron-impurity coupling $\Gamma(\epsilon)=\rho(\epsilon)t^2(\epsilon)$ is energy independent. In the renormalization group approach, the physics of the system is assumed to be governed by $\Gamma(\epsilon)$ only rather than by separate forms of $\rho(\epsilon)$ and $t(\epsilon)$. However, an exact Bethe Ansatz solution of the toy Anderson model demonstrates a crucial role of a form of inverse band dispersion $k(\epsilon)$.