Higher-order, generically complete, continuous, and polynomial-time invariants of crystalline materials

Abstract This paper is at the interface between applied mathematics and numerical computations in materials science. The still open fundamental problem is to quantify the structural novelty of any solid crystalline material (briefly, a periodic crystal) whose atoms can be considered zero-sized points with or without atomic types. Several high-profile papers in Nature reported materials that turned out to be noisy disguises or near-duplicates of already existing structures. We introduce a new mathematical method to quickly and reliably find nearest neighbors of any newly claimed periodic crystal within available databases. The proposed structural invariants are strictly stronger than past descriptors satisfying the same conditions of polynomial-time computability and continuity under noise. The stability and asymptotic of new algorithms are rigorously analyzed.