Duality and the modular group in the quantum Hall effect

Abstract

We explore the consequences of introducing a complex conductivity into the quantum Hall effect. This leads naturally to an action of the modular group on the upper-half complex conductivity plane. Assuming that the action of a certain subgroup, compatible with the law of corresponding states, commutes with the renormalisation group flow, we derive many properties of both the integer and fractional quantum Hall effects including: universality; the selection rule |p1q2 − p2q1| = 1 for transitions between quantum Hall states characterised by filling factors ν1 = p1/q1 and ν2 = p2/q2; critical values of the conductivity tensor; and Farey sequences of transitions. Extra assumptions about the form of the renormalisation group flow lead to the semi-circle rule for transitions between Hall plateaux.

Dualities in Quantum Hall System and Noncommutative Chern-Simons Theory

“…Many properties of the QHE system just follows from the consistency of the RG flows with the duality group. For instance the semi-circle law [26], universality of the transition conductivities as well as selection rules for the transitions between plateaux [27] follow from the duality property. The universal critical points which map to themselves under the duality group are predicted.…”

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

“…Second, our derivation helps clarify the assumptions which underlie analyses of the consequences of Γ 0 (2) invariance for the renormalization-group (RG) flow in the σ xx − σ xy plane, since we show that this only relies on the underlying particle-vortex duality and on the long-wavelength limit. This is important because it has been shown [13][14][15] that most of the unique features of Quantum Hall electromagnetic response follow from the consistency of Γ 0 (2) invariance with RG flow in the σ xx − σ xy plane, independent of the detailed form of the flow's β-function. (The constraints on β which follow from this symmetry have also been considerably explored [14,[16][17][18][19].)…”

“…This is important because it has been shown [13][14][15] that most of the unique features of Quantum Hall electromagnetic response follow from the consistency of Γ 0 (2) invariance with RG flow in the σ xx − σ xy plane, independent of the detailed form of the flow's β-function. (The constraints on β which follow from this symmetry have also been considerably explored [14,[16][17][18][19].) Previously the key assumption of twodimensional flow, governed by Γ 0 (2) invariance, was just that: an assumption, although a plausible one motivated by analogy with the two-dimensional scaling theory of disorder [20][21][22][23].…”

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

“…The derivation of the Γ 0 (2) action in [17] required performing a Gaussian integral about a fixed background, different backgrounds give different initial conductivities, but in each case the dynamics contributing to the fluctuations are the same and the dynamics of the final system have the same form as that of the initial one. This motivates the suggestion [84,85,86,87] that the scaling flow, which is governed by the fluctuations, should commute with Γ 0 (2), i.e. although Γ 0 (2) is not a symmetry of all the physics it is a symmetry of the scaling flow.…”

“…Firstly one can show that any fixed point of Γ 0 (2), with σ xx > 0, must be fixed point of the scaling flow (though not vice versa), [84]. To see this assume that the scaling flow commutes with the action of Γ 0 (2) and let σ * be a fixed point of Γ 0 (2), i.e.…”