This paper shows how a standard tool for one‑dimensional quantum systems, called matrix product states (MPS), can be adapted to treat symmetries that change from site to site. Such “modulated” symmetries act differently at different places on a chain. The authors derive a revised rule for how those symmetry actions pass through an MPS tensor and use that rule to classify symmetry‑protected topological (SPT) phases and to state Lieb–Schultz–Mattis (LSM)‑type constraints for these systems.

At a high level, an MPS represents a quantum state by tensors on each site and by virtual degrees of freedom on the bonds between sites. For ordinary, uniform symmetries one can “push” the symmetry action from the physical site to the virtual bonds and read off how the edges transform. The paper shows that for modulated symmetries the pushed‑through action becomes site dependent: the symmetry on site j is absorbed into two possibly different unitaries on the left and right virtual bonds of that site. Translation symmetry then links these bond unitaries from site to site. The authors express the allowed projective (that is, up to a phase) ways the virtual bonds can transform by conditions on mathematical objects called cocycles. When those consistency conditions fail, no injective (i.e., generic, nondegenerate gapped) MPS can realize the symmetry assignments, producing an LSM‑type obstruction.

This matters because modulated symmetries appear in several physical contexts, for example in conservation laws that depend on position, in some quantum Hall settings, and in strongly tilted optical lattices. By putting these cases into the MPS language, the work gives a controlled way to list which 1D SPT phases are possible under a given spatial modulation. It also gives a unified route to say when a simple, unique gapped ground state is impossible and the system must either break symmetry, be gapless, or have more exotic order.