I am wondering if it makes any sense to non-minimally (say, Pauli-like) couple an external gauge field with a non-relativistic scalar field: \begin{equation} p_\mu \rightarrow p_\mu - e A_\mu + \epsilon_{\mu \nu \rho}F^{\nu \rho} , \end{equation}
here $F^{\nu \rho}$ is the electromagnetic tensor, written for two spatial dimensions (note that I am not referring to the gravitational gauge field). Of course, one may calculate some observables in such a scenario, however, does there exist any experimental paradigm where such calculations can be verified? I presume that, if such couplings can be realized, many important effects related to the breaking of time-reversal symmetry can be realized without applying any magnetic field.