Geometric frustration inherent to the trillium lattice, a sublattice of the B20 structure

We study the classical Heisenberg model on a recently identified three dimensional corner-shared equilateral triangular lattice, a magnetic sublattice to a large class of systems with the symmetry group $P{2}_{1}3$. Since the degree of geometric frustration of the nearest neighbor antiferromagnetic model on this lattice lies on the border between the pyrochlore (not ordered) and hexagonal (ordered) lattices, it is nontrivial to predict its ground state. Using a classical rotor model, we find an ordered ground state with wave vector $(\frac{2\ensuremath{\pi}}{3{a}_{0}},0,0)$ featuring 120\ifmmode^\circ\else\textdegree\fi{} rotated spins on each triangle. However, a mean field approximation on this lattice fails to find an ordered ground state, finding instead a nontrivially degenerate ground state. As the mean field approach is known to agree with Monte Carlo on the pyrochlore lattice, the reasons for this discrepancy are discussed. We also discuss the possible relevance of our results to MnSi.