Representation-Dependent Machine Learning of the Isotropic-Nematic Transition in the Lebwohl-Lasher Model

What the math gives to ML

The paper identifies a transferable failure mode: unsupervised representation learning can mistake symmetry-related configurations for distinct states when the input space is not quotiented by the physical symmetry group. For apolar orientational variables, global rotations and sign reversals preserve the physics but can produce large distances in raw coordinate space. The transferable construction is a local-correlation representation based on the even second Legendre polynomial used by the model Hamiltonian. In neural networks, this can become an invariant input layer or contrastive consistency constraint, with the falsifiable prediction that latent distributions become phase-sensitive and bimodal near coexistence.