From Paramagnet to Dipolar Topological Order via Duality and Dipolar SPT

Abstract: A scheme for the adaptive preparation of a topological state with dipole symmetry, dubbed the dipolar topological state (dTS), which serves as an example of translation symmetry-enriched topological phase, is proposed. The midcircuit state emerging during the preparation process is identified as a two-dimensional symmetry-protected topological (SPT) state protected by dipole bundle symmetry alongside charge and 1-form symmetries. The non-trivial boundary modes of the dipolar SPT state exhibiting the spontaneous breaking of charge and dipole bundle symmetries are analyzed. The duality map between the paramagnetic state and the dipolar topological state is established in the framework of the {\it simultaneous gauging} of two charge symmetries and one dipole symmetry that cannot be reduced as sequential gauging of the individual symmetry. Leveraging this duality, we work out the phase diagram of the dipolar topological state under perturbations by various transverse fields.