Evidence for charge-vortex duality at the LaAlO$_3$-SrTiO$_3$ interface*

The conducting gas formed at the inteface between LaAlO$_3$ and SrTiO$_3$ undergoes a superconductor to insulator transition (SIT) on the application of a back gate voltage, $V_g$. The system also shows evidence of ferromagnetic order coexisting with superconductivity.\footnote{Dikin \textit{et al}, PRL \textbf{107}, 056802 (2011)} The juxtaposition of the ferromagnet with the conducting gas allows for the observation of a novel manifestation of charge-vortex duality. The field due to the magnetization dynamics in the ferromagnet causes a sharp \textit{increase} in resistance on the superconducting side of the transition, in the magnetoresistance measurements, and a sharp \textit{decrease} in resistance on the insulating side. The system can be modeled as an array of Josephson junctions, with two characteristic energy scales: a Josephson coupling energy $E_J$, and a Coulomb charging energy $E_c$. $V_g$ then tunes the ratio, $E_J/E_c$, to cause the transition. We will present external field sweep-rate dependent magnetoresistance data on both sides of the transition to elucidate the nature of the superconducting and insulating states.