# Phase boundaries of the pseudogap Anderson and Kondo models

## TL;DR

This paper investigates phase boundaries in pseudogap Anderson and Kondo models with power-law density of states, revealing different quantum phase transition types and validating poor man's scaling against numerical renormalization group results.

## Contribution

It provides a detailed analysis of phase boundaries and quantum critical points in pseudogap impurity models using poor man's scaling, including cases with diverging density of states.

## Key findings

- Identifies phase boundaries for $0<r<1/2$ and $r>1$ in the Anderson model.
- Derives phase boundaries and scaling trajectories for particle-hole-symmetric Kondo models.
- Shows good agreement between poor man's scaling and numerical renormalization group results.

## Abstract

We use the poor man's scaling approach to study the phase boundaries of a pair of quantum impurity models featuring a power-law density of states $\rho(\omega)\propto|\omega|^r$ that gives rise to quantum phase transitions between local-moment and Kondo-screened phases. For the Anderson model with a pseudogap (i.e., $r>0$), we find the phase boundary for (a) $0<r<1/2$, a range over which the model exhibits interacting quantum critical points both at and away from particle-hole symmetry, and (b) $r>1$, where the phases are separated by first-order quantum phase transitions. For the particle-hole-symmetric Kondo model with easy-axis or easy-plane anisotropy of the spin exchange, the phase boundary and scaling trajectories are obtained for both $r>0$ and $r<0$ (the later case describing a density of states that diverges at the Fermi energy). Comparison with nonperturbative results from the numerical renormalization group shows that poor man's scaling correctly describes the shape of phase boundaries expressed as functional relations between model parameters.