Fractons in Product Codes

We study fracton order in quantum codes obtained as products of classical low density parity check (LDPC) codes. Product codes include as special cases such models as fractal spin liquids on cubic lattices and a lineon model on a general graph.

We propose conditions applicable to the classical codes which lead the product quantum codes to have fracton order. In the case of hypergraph products, satisfaction of these conditions by one or two codes gives rise to type-I and type-II fractons, respectively. By studying the typicality of satisfaction of these conditions for in randomized LDPC constructions, we establish product codes as a natural setting for fracton order. We propose a general relation between fracton order and product codes, with emphasis on new families of fractons.