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We study the magnetic phases of a non-equilibrium spin chain, where coherentinteractions between neighboring lattice sites compete with alternating gainand loss processes. This competition between coherent and incoherent dynamicsinduces transitions between magnetically aligned and highly mixed phases,across which the system changes from a low- to an effectiveinfinite-temperature state. We show that the origin of these transitions can betraced back to the dynamical effect of parity-time-reversal symmetry breaking,which has no counterpart in the theory of equilibrium phase transitions. Thismechanism also results in very atypical features and we find first-ordertransitions without phase co-existence and mixed-order transitions which do notbreak the underlying $U(1)$ symmetry, even in the appropriate thermodynamiclimit. Thus, despite its simplicity, the current model considerably extends thephenomenology of non-equilibrium phase transitions beyond that commonly assumedfor driven-dissipative spins and related systems.

Non-equilibrium magnetic phases in spin lattices with gain and loss