  The dynamical behaviour of many-body systems is often richer than what can be
anticipated from their static properties. Here we show that in closed quantum
systems this becomes evident by considering time-integrated observables as
order parameters. In particular, the analytic properties of their generating
functions, as estimated by full-counting statistics, allow to identify
dynamical phases, i.e. phases with specific fluctuation properties of
time-integrated observables, and to locate the transitions between these
phases. We discuss in detail the case of the quantum Ising chain in a
transverse field. We show that this model displays a continuum of quantum
dynamical transitions, of which the static transition is just an end point.
These singularities are not a consequence of particular choices of initial
conditions or other external non-equilibrium protocols such as quenches in
coupling constants. They can be probed generically through quantum jump
statistics of an associated open problem, and for the case of the quantum Ising
chain we outline a possible experimental realisation of this scheme by digital
quantum simulation with cold ions.
