The Berezinskii-Kosterlitz-Thouless (BKT) phase transition drives the unbinding of topological defects in many two-dimensional systems. In the two-dimensional Coulomb gas, it corresponds to an insulator-conductor transition driven by charge deconfinement. We investigate the global topological properties of this transition, both analytically and by numerical simulation, using a lattice-field description of the two-dimensional Coulomb gas on a torus. The BKT transition is shown to be an ergodicity breaking between the topological sectors of the electric field, which implies a definition of topological order in terms of broken ergodicity. The breakdown of local topological order at the BKT transition leads to the excitation of global topological defects in the electric field, corresponding to different topological sectors. The quantized nature of these classical excitations, and their strict suppression by ergodicity breaking in the low-temperature phase, afford striking global signatures of topological-sector fluctuations at the BKT transition. We discuss how these signatures could be detected in experiments on, for example, magnetic films and cold-atom systems.

We apply the irreversible event-chain Monte Carlo (ECMC) algorithm to the simulation of dense all-atom systems with long-range Coulomb interactions. ECMC is event-driven and exactly samples the Boltzmann distribution. It neither uses time-step approximations nor spatial cutoffs on the range of the interaction potentials. Most importantly, it need not evaluate the total Coulomb potential and thus circumvents the major computational bottleneck of traditional approaches. It only requires the derivatives of the two-particle Coulomb potential, for which we discuss mutually consistent choices. ECMC breaks up the total interaction potential into factors. For particle systems made up of neutral dipolar molecules, we demonstrate the superior performance of dipole-dipole factors that do not decompose the Coulomb potential beyond the two-molecule level. We demonstrate that these long-range factors can nevertheless lead to local lifting schemes, where subsequently moved particles are mostly close to each other. For the simple point-charge water model with flexible molecules (SPC/Fw), which combines the long-ranged intermolecular Coulomb potential with hydrogen-oxygen bond-length vibrations, a flexible hydrogen-oxygen-hydrogen bond angle, and Lennard-Jones oxygen-oxygen potentials, we break up the potential into factors containing between two and six particles. For this all-atom liquid-water model, we demonstrate that the computational complexity of ECMC scales very well with the system size. This is achieved in a pure particle-particle framework, without the interpolating mesh required for the efficient implementation of other modern Coulomb algorithms. Finally, we discuss prospects and challenges for ECMC and outline several future applications.