Emergent electrostatics in planar XY spin models: The bridge connecting topological order with broken U(1) symmetry

Topological phases have been a central focus of condensed-matter physics for over 50 years.  Along with many experimental applications, they have provided much intellectual interest due to their characterization via some form of topological ordering, as opposed to the symmetry-breaking ordering of conventional continuous phase transitions.  This distinction is most subtle in the case of the Berezinskii—Kosterlitz—Thouless (BKT) transition as its experimental realizations appear to break U(1) symmetry at low temperature.  It also presents two further paradoxes: i) its prototypical short-range interacting planar XY spin model behaves as an emergent long-range interacting electrolyte; ii) its topological ordering is not accompanied by a topological nonergodicity within the BKT picture.  This data deposit supports a review paper that addresses these three interconnected questions.  In particular, it provides the simulation data that supports the concept of topological nonergodicity throughout the low-temperature phase.  Simulation data that supports the concept of broken U(1) symmetry (at low temperature) is reproduced from another research article.  This data was stored in the University of Bristol data repository, data.bris, with DOI 10.5523/bris.3ov1rl6xtshwv2iuixrbs6f39q.  Simulation data used to demonstrate low-temperature broken symmetry in the two-dimensional Ising model is also provided in the present data deposit.