  The sub-ohmic spin-boson model is known to possess a novel quantum phase
transition at zero temperature between a localised and delocalised phase. We
present here an analytical theory based on a variational ansatz for the ground
state, which describes a continuous localization transition with mean-field
exponents for $0<s<0.5$. Our results for the critical properties show good
quantitiative agreement with previous numerical results, and we present a
detailed description of all the spin observables as the system passes through
the transition. Analysing the ansatz itself, we give an intuitive microscopic
description of the transition in terms of the changing correlations between the
system and bath, and show that it is always accompanied by a divergence of the
low-frequency boson occupations. The possible relevance of this divergence for
some numerical approaches to this problem is discussed and illustrated by
looking at the ground state obtained using density matrix renormalisation group
methods.
