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We use ab initio electronic structure calculations in combination with MonteCarlo simulations to investigate the magnetic and ferroelectric properties ofbulk orthorhombic HoMnO$_3$ and ErMnO$_3$. Our goals are to explain theinconsistencies in the measured magnetic properties of the orthorhombicperovskite manganites (o-$R$MnO$_3$) with small rare-earth ($R$) cations or Y,as well as the contradictions between the directions and amplitudes of theelectric polarizations reported by different experimental groups. Ourcomputations stabilize several exotic magnetic orders (so-called w-spiral,H-AFM and I-AFM), whose presence resolve the contradictions in the measuredmagnetic and ferroelectric properties of o-$R$MnO$_3$. We show that theseorders emerge due to strong four-spin ring exchange interactions.

Four-spin ring interaction as a source of unconventional magnetic orders in orthorhombic perovskite manganites