Particle-vortex duality and the modular group:  Applications to the quantum Hall effect and other two-dimensional systems

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.

Global Phase Diagram of the High-Tc Cuprates

“…In the case of the QHE, the "Law of Corresponding States" physically relates all quantum phase transitions at various filling fractions to a single quantum phase transition from the ν = 1 integer state to the Hall insulator. In recent years, this powerful mapping among the different fractional states has been made more precise by the derivation of the SL(2, Z) discrete modular group transformation from the Chern-Simons theory 45,46,47 . Similarly, the central idea of the current paper is to relate the fractional Mott insulator to SC transition with the transition from the AF Mott insulator at half-filling to SC state, which is already well understood within the context of the original, simple SO(5) theory.…”

Modular Symmetry and Fractional Charges in N = 2 Supersymmetric Yang-Mills and the Quantum Hall Effect

“…• Exact flow lines in the σ plane are immediate consequences of Γ θ invariance and particlehole symmetry, [81]. The results are again semi-circles or vertical lines in the σ plane, implying a semi-circle law for these bosonic systems.…”

“…2-dimensional superconductors, one can obtain predictions from the fermionic case by using the flux attachment transformation to add an odd number of vortices to the quasi-particles. For the case of a single unit of flux this is equivalent to conjugating Γ 0 (2) by F = S −1 T S. Since F −1 F 2 F = F 2 and F −1 T F = F −2 S the resulting group is generated by S and F 2 , or equivalently S and T 2 , and is the group Γ θ , [2,82,81]. Note that the S transformation in (4.22) requires ϑ = −s = ±η with η → ∞ in (4.19).…”

“…Γ 0 (2) is the subgroup of the full modular group that is relevant for systems with fermionic charge carriers in a strong perpendicular magnetic field with the spins well split. Building on the work of [17], and following a suggestion in [114], it was shown in [81] that Γ 0 (2) should be the group relevant to 2-dimensional systems involving spin polarised fermionic quasi-particles while Wilczek and Shapere's group Γ θ should be relevant to 2-dimensional systems involving bosonic quasi-particles. If the effective charge carriers are bosonic, e.g.…”

“…The analysis of was taken further in [81] and extended beyond the regime of linear response in [82].…”