Highly controllable switching pathways in multiferroic GdMn$_2$O$_5$

Controlling magnetic moments using electric fields remains a central challenge in spintronics. Multiferroics, where magnetic and electric orders coexist, may be a natural platform for such control, but progress has been limited because interactions between these orders are typically too weak to overcome the energy barriers between magnetic states. A recently demonstrated topologically protected switching circumvents this limitation by exploiting reduced barriers at a phase transition. Nevertheless, the conditions enabling this phenomenon remain elusive and electric field control is poorly understood. Here, we experimentally map the energy landscape by tracking transitions in GdMn$_2$O$_5$ under combined electric and magnetic fields. The experiments reveal that the switching pathways can be controlled by the electric field. A minimal theoretical model captures the observed behavior, identifies the parameter space where switching paths are sensitive to small perturbations and reveals design principles for implementing topological switching in other materials.