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

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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 P2₁3. 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 non-trivial to predict its ground state. Using a classical rotor model, we find an ordered ground state with wavevector (2π/3a₀,0,0) featuring 120ᵒ rotated spins on each triangle. However, a mean field approximation on this lattice fails to find an ordered ground state, finding instead a non-trivially 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.