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Physical Review Letters

Universality of Quantum Hall Effect: Topological Invariant and Observable

Hideo Aoki, Tsuneya Ando

Published Dec 15, 1986• 50 citations
Publisher

Abstract

We show a universality in the quantum Hall effect in that the flux-averaged Hall conductivity in any two- or three-dimensional system with or without disorder is given by a topological invariant quantized in units of $\frac{{e}^{2}}{h}$ at $T=0$ for every energy level in a finite system. Relevance to the observable Hall conductivity over the whole energy spectrum is presented together with numerical results for lattice systems.

DOI
10.1103/physrevlett.57.3093

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We show a universality in the quantum Hall effect in that the flux-averaged Hall conductivity in any two- or three-dimensional system with or without disorder is given by a topological invariant quantized in units of e2h\frac{{e}^{2}}{h}he2​ at T=0T=0T=0 for every energy level in a finite system. Relevance to the observable Hall conductivity over the whole energy spectrum is presented together with numerical results for lattice systems.
I consider a gas of ``free'' anyons with statistical parameter \ensuremath{\delta}, with hard cores, on a two-dimensional square lattice. Using a recently derived Jordan-Wigner transformation, I map this problem onto a gas of fermions on the same lattice coupled to a Chern-Simons gauge theory with coupling \ensuremath{\theta}=1/2\ensuremath{\delta}. At the semiclassical level, the system is found to be equivalent to a gas of fermions, with the same density, in an average effective magnetic field \ensuremath{\rho}/\ensuremath{\theta}. I consider the case in which an integer number of the Landau bands of the saddle-point problem are completely filled. If \ensuremath{\delta}=\ensuremath{\pi}/m and the density \ensuremath{\rho}=r/q, with m, r, and q integers, the system is a superfluid, provided that q is larger than twice the largest common factor of m and r. If q is even and the system is half filled, the state may be either a superfluid or a quantum Hall phase. For all other values of \ensuremath{\rho} and \ensuremath{\delta}, compatible with integer filling of the Landau bands, the system is in a quantum Hall phase. The dynamical stability of the superfluid state is ensured by the topological invariance of the quantized Hall conductance of the fermion problem. I find a close analogy between anyon superconductivity and the Schwinger mechanism. The effective Lagrangian for the low-energy modes coupled to the electromagnetic field is derived. The energies of fermion and flux states are logarithmically divergent, but finite for the anyon state. The system has flux quantization, a zero-temperature Hall effect with a quantized Hall conductance, Meissner effect, charged vortices, screening with induced magnetic fields for static charges, and different masses for the longitudinal and transverse components of the electromagnetic field.
The Haldane model is a standard tight-binding model describing electrons hopping on a hexagonal lattice subject to a transverse, dipolar magnetic field. We consider its interacting version for values of the interaction strength that are small compared to the bandwidth. We study the critical case at the transition between the trivial and ``topological'' insulating phases, and we rigorously establish that the transverse conductivity on the dressed critical line is quantized at a half-integer multiple of e2/h{e}^{2}/he2/h: this is the average of the integer values of the Hall conductivity in the insulating phases on either side of the dressed critical line. Together with previous results, this fully characterizes the nature of the phase transition between different Hall plateaus and proves its universality with respect to many-body interactions. The proof is based on a combination of constructive renormalization group methods and exact lattice Ward identities.
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