Dissipative Dynamical Phase Transition as a Complex Ising Model

It is expected that generic open quantum systems, in the absence of monitoring by an observer, are driven towards a featureless infinite-temperature state due to decoherence by the environment. In this work, we study a (1+1)-d toy model with mixed unitary and decohering dynamics and show that it is nevertheless possible to see interesting dynamical phase transitions in the approach to the steady state, as probed by the expectation values of non-local observables. Our model is described by a quantum Ising model with imaginary transverse field, which hosts a gapped ferromagnetic phase as well as a gapless algebraically-ordered phase. These phases meet at an unusual "two-sided" transition which has properties similar to both first- and second-order transitions. Finally, we show how the unusual features of this model can be accessed through simple protocols on a quantum simulator.