Lattice symmetries and regular states in classical frustrated antiferromagnets

We consider some classical and frustrated lattice spin models with global O(3) spin symmetry. There is no general analytical method to find a ground-state if the spin dependence of the Hamiltonian is more than quadratic (i.e. beyond the Heisenberg model) and/or if the lattice has more than one site per unit cell. To deal with these situations, we introduce a family of variational spin configurations, dubbed "regular states", which respect all the lattice symmetries modulo global O(3) spin transformations (rotations and/or spin flips). The construction of these states is explicited through a group theoretical approach, and all the regular states on the square, triangular, honeycomb and kagome lattices are listed. Their equal time structure factors and powder-averages are shown for comparison with experiments. All the well known N\'eel states with 2 or 3 sublattices appear amongst regular states on various lattices, but the regular states also encompass exotic non-planar states with cubic, tetrahedral or cuboctahedral geometry of the T=0 order parameter. Whatever the details of the Hamiltonian (with the same symmetry group), a large fraction of these regular states are energetically stationary with respect to small deviations of the spins. In fact these regular states appear as exact ground-states in a very large range of parameter space of the simplest models that we have been looking at. As examples, we display the variational phase diagrams of the J1-J2-J3 Heisenberg model on all the previous lattices as well as that of the J1-J2-K ring-exchange model on square and triangular lattices.