Geometrical Frustration Beyond Magnets*

Geometrical frustration – the inability of a system to satisfy all of its interactions simultaneously because of geometrical constraints – can suppress conventional ordering and promote the formation of exotic states that are disordered, yet strongly correlated. Materials in which magnetic spins occupy lattices built from corner or edge-sharing triangles have provided many examples of novel magnetic behavior due to frustration. In this talk, I will discuss how frustration of structural (nonmagnetic) degrees of freedom – charge states, orbital orientations, or chain displacements – can determine the structures and properties of materials. I will also discuss how the nonmagnetic frustrated interactions can be mapped to equivalent “toy” spin Hamiltonians, including those that are challenging to realize experimentally in magnets. I illustrate these points using three real examples of nonmagnetic frustration. First, I discuss how the solid phases of silver(I) and/or gold(I) cyanides, in which polymeric chains occupy a triangular lattice, can host structural analogs of the spin vortices of triangular XY magnets [1]. Second, I discuss the pyrochlore oxide Y₂Mo₂O₇, and explain how orbital dimerization of Jahn-Teller active Mo⁴⁺ ions on the frustrated pyrochlore lattice may yield an orbital-ice analog of spin-ice and water-ice states [2]. Finally, I discuss the triangular-lattice-based system YbMgGaO₄ [3,4] – of interest because of proposed quantum-spin-liquid-like behaviour of its magnetic Yb³⁺ ions – and show how the charge difference between nonmagnetic Mg²⁺and Ga³⁺generates a structurally-frustrated state, with implications for the proposed quantum-spin-liquid behaviour.

[1] Cairns, Cliffe, Paddison et al., Nat Chem 8, 442 (2016).
[2] Thygesen, Paddison, Zhang et al., PRL 118, 067201 (2017).
[3] Li, Chen, Tong et al., PRL 115, 167203 (2015).
[4] Paddison, Daum, Dun et al., Nat Phys 13, 117 (2017).