Global symmetries of quantum Hall systems: Lattice description

I analyze nonlocal symmetries of finite-size Euclidean three-dimensional lattice Chern-Simons models in the presence of an external magnetic field and nonzero average current. It is shown that under very general assumptions the particle-vortex duality interchanges the total Euclidean magnetic flux $\mathbf{\Phi }/2\mathrm{}\pi$ with the total current I in a given direction, while the flux attachment transformation increases the flux in a given direction by the corresponding component of the current, $\mathbf{\Phi }/2\mathrm{}\pi \to \mathbf{\Phi }/2\mathrm{}\pi +2\mathrm{}\mathbf{I},$ independently of the disorder. In the language of $2+1$ dimensional models, appropriate for describing quantum Hall systems, these transformations are equivalent to the symmetries of the phase diagram known as the correspondence laws, and the nonlinear current-voltage mapping between mutually dual points, recently observed near the quantum Hall liquid-insulator transitions.