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Isotropic but otherwise largely arbitrary Heisenberg models in the presenceof a homogeneous magnetic field are considered, including various integrable,non-integrable, as well as disordered examples, and not necessarily restrictedto one dimension or short-range interactions. Taking for granted that thenon-equilibrium initial condition and the spectrum of the field-free modelsatisfy some very weak requirements, expectation values of generic observablesare analytically shown to exhibit permanent long-time oscillations, thus rulingout equilibration. If the model (but not necessarily the initial condition) istranslationally invariant, the long-time oscillations are moreover shown toexhibit synchronization in the long run, meaning that they are invariant underarbitrary translations of the observable. Analogous long-time oscillations arealso recovered for temporal correlation functions when the system is already atthermal equilibrium from the outset, thus realizing a so-called time crystal.

Non-equilibration, synchronization, and time crystals in isotropic Heisenberg models