Homological Invariants in Translation-Invariant Pauli Stabilizer Codes

Pauli stabilizer codes serve as foundational models in quantum error correction and the study of exotic quantum phases. In this talk, we explore the application of homological methods to translationally invariant Pauli stabilizer codes with qudits of varying dimensions. We introduce a series of invariants, termed charge modules, and delve into their properties and physical interpretations. A key focus is on codes whose charge modules exhibit zero Krull dimension, a condition indicative of the mobility of excitations. Notably, we demonstrate that this condition is universally met in two-dimensional codes with a unique ground state in infinite volume, extending prior findings beyond the realm of uniform, prime qudit dimensions. For systems where all excitations are mobile, we establish the existence of p-dimensional excitations and associated (D−p−1)-form symmetries corresponding to each element of the p-th charge module. Additionally, we define a braiding pairing between charge modules in complementary degrees.