Q-tensor continuum energies as limits of head-to-tail symmetric spin systems

We consider a class of spin-type discrete systems and analyze their continuum limit as the lattice spacing goes to zero. Under standard coerciveness and growth assumptions together with an additional head-to-tail symmetry condition, we observe that this limit can be conveniently written as a functional in the space of Q-tensors. We further characterize the limit energy density in several cases (both in two and three dimensions). In the planar case we also develop a second-order theory and we derive gradient or concentration-type models according to the chosen scaling.