  We examine the generating function of the time-integrated energy for the
one-dimensional Glauber-Ising model. At long times, the generating function
takes on a large-deviation form and the associated cumulant generating function
has singularities corresponding to continuous trajec- tory (or "space-time")
phase transitions between paramagnetic trajectories and ferromagnetically or
anti-ferromagnetically ordered trajectories. In the thermodynamic limit, the
singularities make up a whole curve of critical points in the complex plane of
the counting field. We evaluate analyti- cally the generating function by
mapping the generator of the biased dynamics to a non-Hermitian Hamiltonian of
an associated quantum spin chain. We relate the trajectory phase transitions to
the high-order cumulants of the time-integrated energy which we use to extract
the dynamical Lee-Yang zeros of the generating function. This approach offers
the possibility to detect continuous trajectory phase transitions from the
finite-time behaviour of measurable quantities.
