Scaling theory of the integer quantum Hall effect

Abstract

The scaling theory of the transitions between plateaus of the Hall conductivity in the integer Quantum Hall effect is reviewed. In the model of twodimensional noninteracting electrons in strong magnetic fields the transitions are disorder-induced localization-delocalization transitions. While experimental and analytical approaches are surveyed, the main emphasis is on numerical studies, which successfully describe the experiments. The theoretical models for disordered systems are described in detail. An overview of the finitesize scaling theory and its relation to Anderson localization is given. The field-theoretical approach to the localization problem is outlined. Numerical methods for the calculation of scaling quantities, in particular the localization length, are detailed. The properties of local observables at the localizationdelocalization transition are discussed in terms of multifractal measures. Finally, the results of extensive numerical investigations are compared with experimental findings.

Robust optical delay lines with topological protection

“…This photonic system not only enables investigations of quantum Hall physics by simulating different types of Hamiltonians at room temperature, but it also taps into topological features to provide robust devices for photonics. In the non-interacting regime (which was the topic of this article), one can explore the Hofstadter butterfly of photons, and photonic edge states as delay lines immune to disorders and also localization in 2D for non-interacting particles [41]. Furthermore, with the addition of interaction between photons, this system opens up exciting prospects for exploring many-body topological state of light.…”

Disordered Dirac fermions: the marriage of three different approaches

“…It is enough to ensure this property at the two singular points z = 0 and z = 1. The blocks F (1) i (z) are regular at z = 0 and logarithmic at z = 1, whilst the blocks F (2) i (z) are logarithmic at z = 0 and regular at z = 1. It is straightforward to see that in this subspace of functions one can only have 10…”

“…The γ µ matrices form a two-dimensional representation of the Clifford algebra {γ µ , γ ν } = 2g µν with Euclidean metric g µν = diag (1,1), and the gauge field A αβ µ = A a µ τ αβ a may be expanded in terms of the generators τ a of su(N C ). Physical quantities are obtained by disorder averaging products of Green's functions.…”

Pumping conductance, the intrinsic anomalous Hall effect, and statistics of topological invariants

“…Instead there are localized regions with occasional isolated energies supporting extended states of nonzero Chern number, and there is strong numerical evidence 11,12 that the localization length diverges near these critical energies as…”

“…It is known that if the disorder strength is kept constant as the system increases in size, eventually "quantum Hall plateau transitions" develop 11,12 . (At very large disorder, all states become localized 13 and there is only the ordinary insulator.)…”