  The quenching of degenerate impurity states in metals generally induces a
long-range correlated quantum state known as the Kondo screening cloud. While a
macroscopic number of particles clearly take part in forming this extended
structure, assessing the number of truly entangled degrees of freedom requires
a careful analysis of the relevant many-body wavefunction. For this purpose, we
examine the natural single-particle orbitals that are eigenstates of the
single-particle density (correlation) matrix for the ground state of two
quantum impurity problems: the interacting resonant level model (IRLM) and the
single impurity Anderson model (SIAM). As a simple and general probe for
few-body versus many-body character we consider the rate of exponential decay
of the correlation matrix eigenvalues towards inactive (fully empty or filled)
orbitals. We find that this rate remains large in the physically most relevant
region of parameter space, implying a few-body character. Genuine many-body
correlations emerge only when the Kondo temperature becomes exponentially
small, for instance near a quantum critical point. In addition, we demonstrate
that a simple numerical diagonalization of the few-body problem restricted to
the Fock space of the most correlated orbitals converges exponentially fast
with respect to the number of orbitals, to the true ground state of the IRLM.
We also show that finite size effects drastically affect the correlation
spectrum, shedding light on an apparent paradox arising from previous studies
on short chains.
