Multipole phases in a type of spin ladders with local conserved quantities and generalizations

We study spin ladder models with exact multipole phases, which are traditional spin phases formed by multipole moments. These phases feature non-trivial order with zero magnetization. The multipole models have dimer local conserved quantities that are Ising terms of spin. The Hilbert spaces are locally fragmented into independent sectors described effectively by ``higher-order spin". For dipole models, we consider two ladder geometries with quadratic spin couplings and work out the phase diagrams. Higher-order multipole models are obtained by introducing more dimer conserved quantities. The phases are characterized by the values of the local conserved quantities and the traditional spin phases of the higher-order spin. The dipole models can in principle be realized in experiments, we propose such realization by electric field controlled Rydberg atom arrays.