Superuniversality is the sharing of the same critical exponents among distinct critical points. This surprising behavior is observed at phase transitions between various quantum Hall states found in the two-dimensional electron gas. Motivated by the theoretical challenge posed by superuniversality, we introduce a new effective description that has an emergent $U(N)$ gauge symmetry with $N > 1$ for a quantum phase transition between an integer quantum Hall state and an insulator. We use modular transformations to generate from this theory new effective descriptions for quantum phase transitions between a large class of fractional quantum Hall states whose quasiparticle excitations have abelian statistics. In the 't Hooft large $N$ limit, the correlation length and dynamical critical exponents are independent of the particular transition in the absence of disorder. We argue that this conclusion survives away from the large $N$ limit using recent duality conjectures and, thereby, provide theoretical support for the phenomenon of superuniversality found at distinct quantum Hall critical points.Superuniversality is the sharing of the same critical exponents among distinct critical points. This surprising behavior is observed at phase transitions between various quantum Hall states found in the two-dimensional electron gas. Motivated by the theoretical challenge posed by superuniversality, we introduce a new effective description that has an emergent $U(N)$ gauge symmetry with $N > 1$ for a quantum phase transition between an integer quantum Hall state and an insulator. We use modular transformations to generate from this theory new effective descriptions for quantum phase transitions between a large class of fractional quantum Hall states whose quasiparticle excitations have abelian statistics. In the 't Hooft large $N$ limit, the correlation length and dynamical critical exponents are independent of the particular transition in the absence of disorder. We argue that this conclusion survives away from the large $N$ limit using recent duality conjectures and, thereby, provide theoretical support for the phenomenon of superuniversality found at distinct quantum Hall critical points.

Dec 13, 20173 citations

Integer Quantum Hall Effect for Lattice Fermions

K. Ziegler

A two-dimensional lattice model for non-interacting fermions in a magnetic field with half a flux quantum per plaquette and N levels per site is considered. This is a model which exhibits the integer quantum Hall effect (IQHE) in the presence of disorder. It presents an alternative to the continuous picture for the IQHE with Landau levels. The large- N limit can be solved: two Hall transitions appear and there is an interpolating behaviour between the two Hall plateaux. Although this approach to the IQHE is different from the traditional one with Landau levels because of different symmetries (continuous for Landau levels and discrete here), some characteristic features are reproduced. For instance, the slope of the Hall conductivity is infinite at the transition points and the electronic states are delocalized only at the transitions.A two-dimensional lattice model for non-interacting fermions in a magnetic field with half a flux quantum per plaquette and N levels per site is considered. This is a model which exhibits the integer quantum Hall effect (IQHE) in the presence of disorder. It presents an alternative to the continuous picture for the IQHE with Landau levels. The large- N limit can be solved: two Hall transitions appear and there is an interpolating behaviour between the two Hall plateaux. Although this approach to the IQHE is different from the traditional one with Landau levels because of different symmetries (continuous for Landau levels and discrete here), some characteristic features are reproduced. For instance, the slope of the Hall conductivity is infinite at the transition points and the electronic states are delocalized only at the transitions.

arXiv:cond-mat/9408035Europhysics Letters (EPL)Oct 1, 199410 citations

Inside the quantum Hall effect

Wolfgang Ketterle

Nature PhysicsFeb 1, 20158 citations

Infinite symmetry in the quantum Hall effect

Andrea Cappelli A, Carlo A. Trugenberger, Guillermo R. Zemba

arXiv:hep-th/9206027Nuclear Physics BMay 1, 1993216 citations

Central charge from adiabatic transport of cusp singularities in the quantum Hall effect

Tankut Can

Abstract We study quantum Hall (QH) states on a punctured Riemann sphere. We compute the Berry curvature under adiabatic motion in the moduli space in the large N limit. The Berry curvature is shown to be finite in the large N limit and controlled by the conformal dimension of the cusp singularity, a local property of the mean density. Utilizing exact sum rules obtained from a Ward identity, we show that for the Laughlin wave function, the dimension of a cusp singularity is given by the central charge, a robust geometric response coefficient in the QHE. Thus, adiabatic transport of curvature singularities can be used to determine the central charge of QH states. We also consider the effects of threaded fluxes and spin-deformed wave functions. Finally, we give a closed expression for all moments of the mean density in the integer QH state on a punctured disk.Abstract We study quantum Hall (QH) states on a punctured Riemann sphere. We compute the Berry curvature under adiabatic motion in the moduli space in the large N limit. The Berry curvature is shown to be finite in the large N limit and controlled by the conformal dimension of the cusp singularity, a local property of the mean density. Utilizing exact sum rules obtained from a Ward identity, we show that for the Laughlin wave function, the dimension of a cusp singularity is given by the central charge, a robust geometric response coefficient in the QHE. Thus, adiabatic transport of curvature singularities can be used to determine the central charge of QH states. We also consider the effects of threaded fluxes and spin-deformed wave functions. Finally, we give a closed expression for all moments of the mean density in the integer QH state on a punctured disk.

arXiv:1611.05563Journal of Physics A Mathematical and TheoreticalMar 29, 20178 citations

Superuniversality and non-abelian bosonization in

2+1

dimensions

U(N)

gauge symmetry with

N > 1

for a quantum phase transition between an integer quantum Hall state and an insulator. We use modular transformations to generate from this theory new effective descriptions for quantum phase transitions between a large class of fractional quantum Hall states whose quasiparticle excitations have abelian statistics. In the 't Hooft large

N

limit, the correlation length and dynamical critical exponents are independent of the particular transition in the absence of disorder. We argue that this conclusion survives away from the large

N

limit using recent duality conjectures and, thereby, provide theoretical support for the phenomenon of superuniversality found at distinct quantum Hall critical points.