Matrix Product States for Modulated Topological Phases: Crystalline Equivalence Principle and Lieb-Schultz-Mattis Constraints

Modulated symmetries are internal symmetries that act in a spatially non-uniform manner. Consequently, when a modulated symmetry G_int is combined with a spatial symmetry G_sp, the total symmetry group takes the form of a semidirect product G=G_intrtimes G_sp. Using matrix product states, we classify topological phases protected by modulated symmetries together with spatial symmetries in one spatial dimension. We show that these modulated symmetry-protected topological (SPT) phases are classified by H²(G,U(1)_s), in agreement with the crystalline equivalence principle, which states that SPT phases protected by symmetries involving spatial elements are in one-to-one correspondence with internal SPT phases protected by the same symmetries, viewed as acting internally. Furthermore, we provide a matrix product state derivation of the Lyndon-Hochschild-Serre spectral sequence for the corresponding internal SPT phases, which enables us to construct an explicit correspondence between modulated SPT phases and internal SPT phases. As applications of this classification, we prove a Lieb-Schultz-Mattis (LSM) theorem for modulated symmetries that forbids the existence of symmetric short-ranged entangled ground state, as well as an SPT-LSM constraint that enforces nontrivial entanglement in the SPT ground state. Finally, we use the classification to establish a similar LSM-type constraints for non-invertible Kramers-Wannier reflection symmetries.