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In the dense metal-organic framework Na[Mn(HCOO)$_3$], Mn$^{2+}$ ions($S=\frac{5}{2}$) occupy the nodes of a `trillium' hyperkagome net. We showthat this material exhibits a variety of behaviour characteristic of geometricfrustration: the N\'eel transition is suppressed well below the characteristicmagnetic interaction strength; short-range magnetic order persists far abovethe N\'eel temperature; and the magnetic susceptibility exhibits apseudo-plateau at $\frac{1}{3}$-saturation magnetisation. We demonstrate that asimple nearest-neighbour Heisenberg antiferromagnet model accountsquantitatively for each observation, and hence Na[Mn(HCOO)$_3$] is the firstexperimental realisation of this model on the trillium net. We develop amapping between this trillium model and that on the two-dimensionalShastry-Sutherland lattice, and demonstrate how both link geometric frustrationwithin the classical spin liquid regime to a strong magnetocaloric response atlow fields.

Geometric frustration on the trillium lattice in a magnetic metal-organic framework