  The adiabatic theorem is a fundamental result established in the early days
of quantum mechanics, which states that a system can be kept arbitrarily close
to the instantaneous ground state of its Hamiltonian if the latter varies in
time slowly enough. The theorem has an impressive record of applications
ranging from foundations of quantum field theory to computational recipes in
molecular dynamics. In light of this success it is remarkable that a
practicable quantitative understanding of what "slowly enough" means is limited
to a modest set of systems mostly having a small Hilbert space. Here we show
how this gap can be bridged for a broad natural class of physical systems,
namely many-body systems where a small move in the parameter space induces an
orthogonality catastrophe. In this class, the conditions for adiabaticity are
derived from the scaling properties of the parameter dependent ground state
without a reference to the excitation spectrum. This finding constitutes a
major simplification of a complex problem, which otherwise requires solving
non-autonomous time evolution in a large Hilbert space. We illustrate our
general results by analyzing conditions for the transport quantization in a
topological Thouless pump.
