Ferromagnets and staggered antiferromagnets are perhaps the most common forms of magnetic orderings that we come across in models and materials. However, there are many systems that support unconventional magnetic orderings which are non-collinear and, in some situations, even non-coplanar. In fact, during the last few years non-collinear and non-coplanar magnetic states have proven to be of special interest for condensed matter physicists. A large number of multiferroic compounds, e.g., TbMnO 3 , FeVO 4 , CuO, etc., owe their ferroelectric behavior to the existence of a spiral magnetic order [1]. Anomalous Hall effect observed in metallic pyrochlore compounds, such as Pr 2 Ir 2 O 7 , Nd 2 Mo 2 O 7 , etc. is attributed to the presence of non-coplanar magnetic states in these materials [2]. While the deviation from collinearity in magnetic order is easy to explain in terms of competing exchange interactions between well localized magnetic moments, or geometrical frustrations, the same is not true for non-coplanar magnetism.
We explore the Kondo-lattice model and the Hubbard model on geometrically frustrated lattices for the possibility of non-coplanar magnetic phases. I will begin by describing results for the Kondo-lattice model on triangular lattice, where a non-coplanar magnetic phases is stable at 1/4 and 3/4 fillings [3]. The same model when defined on a checkerboard lattice leads to a different type of non-coplanar magnetic state which supports quantum Hall effect without external magnetic field, and is a realization of Haldane's model [4]. Finally, I will discuss some ongoing work on the possible stability of such non-coplanar phases within a single band Hubbard model on triangular and checkerboard geometries.
[1] Phys. Rev. Lett. 95, 087206 (2005); Phys. Rev. B 80, 220402 (2009); Nature Mater. 7, 291 (2008)
[2] Phys. Rev. Lett. 111, 036602 (2013); Phys. Rev. Lett. 111, 196601 (2013); Phys. Rev. Lett. 90, 257202 (2003)
[3] Phys. Rev. Lett. 101, 156402 (2008); Phys. Rev. Lett. 105, 216405 (2010)
[4] Phys. Rev. Lett. 109, 166405 (2012); Phys. Rev. Lett. 61, 2015 (1988)