  Quantum anomalies offer a useful guide for the exploration of transport
phenomena in topological semimetals. In this work, we introduce a model
describing a semimetal in four spatial dimensions, whose nodal points act like
tensor monopoles in momentum space. This system is shown to exhibit
monopole-to-monopole phase transitions, as signaled by a change in the value of
the topological Dixmier-Douady invariant as well as by the associated surface
states on its boundary. We use this model to reveal an intriguing "4D parity
magnetic effect", which stems from a parity-type anomaly. In this effect,
topological currents are induced upon time-modulating the separation between
the fictitious monopoles in the presence of a magnetic perturbation. Besides
its theoretical implications in both condensed matter and quantum field theory,
the peculiar 4D magnetic effect revealed by our model could be measured by
simulating higher-dimensional semimetals in synthetic matter.
