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

Trajectory phase transitions and dynamical Lee-Yang zeros of the Glauber-Ising chain