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Quantum Hall states of high symmetry

Chetan Nayak, Frank A. Wilczek

arXiv:cond-mat/9501052Nuclear Physics BSep 1, 19956 citations

Infinite symmetry in the quantum Hall effect

C.A. Lütken

The new states of matter and concomitant quantum critical phenomena revealed by the quantum Hall effect appear to be accompanied by an emergent modular symmetry. The extreme rigidity of this infinite symmetry makes it easy to falsify, but two decades of experiments have failed to do so, and the location of quantum critical points predicted by the symmetry is in increasingly accurate agreement with scaling experiments. The symmetry severely constrains the structure of the effective quantum field theory that encodes the low energy limit of quantum electrodynamics of 1010 charges in two dirty dimensions. If this is a non-linear σ-model the target space is a torus, rather than the more familiar sphere. One of the simplest toroidal models gives a critical (correlation length) exponent that agrees with the value obtained from numerical simulations of the quantum Hall effect.The new states of matter and concomitant quantum critical phenomena revealed by the quantum Hall effect appear to be accompanied by an emergent modular symmetry. The extreme rigidity of this infinite symmetry makes it easy to falsify, but two decades of experiments have failed to do so, and the location of quantum critical points predicted by the symmetry is in increasingly accurate agreement with scaling experiments. The symmetry severely constrains the structure of the effective quantum field theory that encodes the low energy limit of quantum electrodynamics of 1010 charges in two dirty dimensions. If this is a non-linear σ-model the target space is a torus, rather than the more familiar sphere. One of the simplest toroidal models gives a critical (correlation length) exponent that agrees with the value obtained from numerical simulations of the quantum Hall effect.

EPJ Web of ConferencesJan 1, 20142 citations

Infinite symmetry in the fractional quantum Hall effect

Michael Flohr, R Varnhagen

We have generalized recent results on the integer quantum Hall effect, constructing explicitly a W 1+ infinity for the fractional quantum Hall effect such that the negative modes annihilate the Laughlin wavefunctions. This generalization has a nice interpretation in Jain's composite-fermion theory. Furthermore, for these models, we have calculated the wavefunctions of the edge excitations, viewing them as area-preserving deformations of an incompressible quantum droplet and have shown that the W 1+ infinity is the underlying symmetry of the edge excitations in the fractional quantum Hall effect. Finally, we have applied this method to more general wavefunctions.We have generalized recent results on the integer quantum Hall effect, constructing explicitly a W 1+ infinity for the fractional quantum Hall effect such that the negative modes annihilate the Laughlin wavefunctions. This generalization has a nice interpretation in Jain's composite-fermion theory. Furthermore, for these models, we have calculated the wavefunctions of the edge excitations, viewing them as area-preserving deformations of an incompressible quantum droplet and have shown that the W 1+ infinity is the underlying symmetry of the edge excitations in the fractional quantum Hall effect. Finally, we have applied this method to more general wavefunctions.

arXiv:hep-th/9309083Journal of Physics A Mathematical and GeneralJun 7, 199428 citations

Inside the quantum Hall effect

Wolfgang Ketterle

Nature PhysicsFeb 1, 20158 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

Universality and phase diagram in the quantum Hall effect

L. W. Wong

Jan 1, 19970 citations

Infinite symmetry in the quantum Hall effect

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