Particle-Vortex Duality and the Modular Group: Applications to the Quantum Hall Effect and Other 2-D Systems

TL;DR

The work identifies particle–vortex duality in two-dimensional systems as generating level-two modular symmetries that act on the complex conductivity $\\sigma =\\sigma_{xy} + i\\sigma_{xx}$. By deriving an effective low-energy action and its dual, the authors obtain explicit transformations for $\\sigma$ and show that fermionic carriers realize the subgroup $\\Gamma_0(2)$ while bosonic carriers realize $\\Gamma_{\\theta}(2)$, constraining RG flows and predicting universal features such as semi-circle flow lines, fixed points, and exponents. The results provide a cohesive, model-independent framework linking Quantum Hall physics and bosonic 2D superconducting systems, with predictions that follow as a single, consistent package once duality holds. They also extend beyond linear response and discuss conditions under which disorder or interactions might spoil the duality, highlighting experimental tests in clean 2D systems and Josephson networks. Overall, the paper reveals a unifying symmetry structure governing EM response in 2D, with potential connections between seemingly disparate phenomena via modular group actions on conductivity.

Abstract

We show how particle-vortex duality implies the existence of a large non-abelian discrete symmetry group which relates the electromagnetic response for dual two-dimensional systems in a magnetic field. For conductors with charge carriers satisfying Fermi statistics (or those related to fermions by the action of the group), the resulting group is known to imply many, if not all, of the remarkable features of Quantum Hall systems. For conductors with boson charge carriers (modulo group transformations) a different group is predicted, implying equally striking implications for the conductivities of these systems, including a super-universality of the critical exponents for conductor/insulator and superconductor/insulator transitions in two dimensions and a hierarchical structure, analogous to that of the quantum Hall effect but different in its details. Our derivation shows how this symmetry emerges at low energies, depending only weakly on the details of dynamics of the underlying systems.