Measurable consequences of the local breakdown of the concept of temperature

Local temperature defined by a local canonical state of the respective subsystem, does not always exist in quantum many-body systems. Here, we give some examples of how this breakdown of the temperature concept on small length scales might be observed in experiments: Measurements of magnetic properties of an anti-ferromagnetic spin-1 chain. We show that those magnetic properties are in fact strictly local. As a consequence, their measurement reveals whether the local (reduced) state can be thermal. If it is, a temperature may be associated to the measurement results, while this would lead to inconsistencies otherwise.