Matrix product state classification of one-dimensional multipole symmetry-protected topological phases

Spatially modulated symmetries are one of the new types of symmetries the symmetry actions of which are position dependent. Yet exotic phases resulting from these spatially modulated symmetries are not fully understood and classified. In this work, we systematically classify one-dimensional bosonic symmetry-protected topological phases of multipole symmetries by employing the matrix product state formalism. The symmetry action induces projective representations at the ends of an open chain, which we identify via group cohomology. In particular, for $r$-pole symmetries---for instance, $r=0$ (global), 1 (dipole), and 2 (quadrupole)---the classification is determined by distinct components of second cohomology groups that encode the boundary projective representations.