Direct observation of collective magnetism at ferroelectric domain walls in multiferroic ErMnO$_{3}$

Multiferroic hexagonal manganities \textit{RE}MnO$_{3}$ (\textit{RE} = Ho, Er, Lu, etc.) have been of great interest because of the coexistence of ferroelectric and magnetic orders. Herein we report cryogenic magnetic force microscopy (MFM) studies of flux-grown ErMnO$_{3}$ single crystals with vortex ferroelectric domain pattern. By uniquely correlating ambient piezo-response force microscopy and low temperature MFM images at the same location, we identified alternating uncompensated magnetic moments at ferroelectric domain walls that correlate over entire vortex network, suggesting collective magnetism at ferroelectric vortex domain walls.