Critical exponent for the quantum Hall transition

Slevin, Keith; Ohtsuki, Tomi

doi:10.1103/physrevb.80.041304

Cited by 159 publications

(200 citation statements)

References 25 publications

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“…We note the following conceptual difference with the usual MacKinnon-Kramer scaling: 15,16 The MacKinnon-Kramer scaling variable is a Lyapunov exponent, which is a nonnegative quantity. Our scaling variable ln |z 0 | changes sign at the phase transition, so it contains information on which side of the transition one is located.…”

Section: Discussion and Relation To The Critical Exponentmentioning

confidence: 99%

“…Here we demonstrate that this approach produces results consistent with earlier calculations based on the scaling of Lyapunov exponents. 16 To make contact with those earlier calculations, we use the same Chalker-Coddington network model 17,18 (rather than the tight-binding model used in the main text). The parameter that controls the plateau transition in the network model is the mixing angle α of the scattering phase shifts at the nodes of the network.…”

Section: Appendix B: Calculation Of the Critical Exponentmentioning

confidence: 99%

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Topological quantum number and critical exponent from conductance fluctuations at the quantum Hall plateau transition

et al. 2011

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The conductance of a two-dimensional electron gas at the transition from one quantum Hall plateau to the next has sample-specific fluctuations as a function of magnetic field and Fermi energy. Here we identify a universal feature of these mesoscopic fluctuations in a Corbino geometry: The amplitude of the magnetoconductance oscillations has an e 2 /h resonance in the transition region, signaling a change in the topological quantum number of the insulating bulk. This resonance provides a signed scaling variable for the critical exponent of the phase transition (distinct from existing positive definite scaling variables).

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Section: Discussion and Relation To The Critical Exponentmentioning

confidence: 99%

Section: Appendix B: Calculation Of the Critical Exponentmentioning

confidence: 99%

Topological quantum number and critical exponent from conductance fluctuations at the quantum Hall plateau transition

et al. 2011

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“…Consideration was based on the understanding that strong spread in p eliminates completely the interference effects, and thus reduces the analysis of the p-q model to the percolation problem, which predicts much smaller localization radius, ξ(p) ∼ 1/p 8/3 , Eq. (52). In order to test this expectation, we incorporated a strong spread in p into transfer-matrix calculation.…”

Section: B Zero Magnetic Fieldmentioning

confidence: 99%

“…Nodes in the of the CC network also have a transparent meaning: they represent saddle points of the random potential, where equipotentials come as close as magnetic length. Various aspects of the Quantum Hall transition, relevant to experiment [43][44][45][46][47][48][49][50][51] , e.g., divergence of the localization radius (scaling 39,52,68 ), critical statistics of energy levels 53 , mesoscopic conductance fluctuations 54,55 , pointcontact conductance 56 , were studied theoretically using the CC model 57 . Testing the levitation scenario microscopically requires to construct a minimal weakly chiral network model, which captures the physics encoded in the system Eq.…”

Section: First Numerical Verificationmentioning

confidence: 99%

Weakly chiral networks and two-dimensional delocalized states in a weak magnetic field

2010

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We study numerically the localization properties of two-dimensional electrons in a weak perpendicular magnetic field. For this purpose we construct weakly chiral network models on the square and triangular lattices. The prime idea is to separate in space the regions with phase action of magnetic field, where it affects interference in course of multiple disorder scattering, and the regions with orbital action of magnetic field, where it bends electron trajectories. In our models, the disorder mixes counter-propagating channels on the links, while scattering matrices at the nodes describe exclusively the bending of electron trajectories. By artificially introducing a strong spread in the scattering strengths on the links (but keeping the average strength constant), we eliminate the interference and reduce the electron propagation over a network to a classical percolation problem. In this limit we establish the form of the disorder -magnetic field phase diagram. This diagram contains the regions with and without edge states, i.e. the regions with zero and quantized Hall conductivities. Taking into account that, for a given disorder, the scattering strength scales as inverse electron energy, we find agreement of our phase diagram with levitation scenario: energy separating the Anderson and quantum Hall insulating phases floats up to infinity upon decreasing magnetic field. From numerical study, based on the analysis of quantum transmission of the network with random phases on the links, we conclude that the positions of the weak-field quantum Hall transitions on the phase diagram are very close to our classical-percolation results. We checked that, in accord with the Pruisken theory, presence or absence of time reversal symmetry on the links has no effect on the line of delocalization transitions. We also find that floating up of delocalized states in energy is accompanied by doubling of the critical exponent of the localization radius. We establish the origin of this doubling within classical-percolation analysis.

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“…The nature of the critical state at and the critical phenomena near the IQH transition are at the focus of intense experimental [8][9][10][11][12][13] and theoretical research. [14][15][16][17][18][19][20][21][22][23] In spite of much effort over several decades, an analytical treatment of most of the critical conducting states in disordered electronic systems, including in particular that of the mentioned IQH transition, has been elusive (although some proposals [14][15][16] have been put forward, but see Refs. 18 and 19).…”

Section: Introductionmentioning

confidence: 99%

Quantum Hall transitions: An exact theory based on conformal restriction

2012

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We revisit the problem of the plateau transition in the integer quantum Hall effect. Here we develop an analytical approach for this transition, and for other two-dimensional disordered systems, based on the theory of "conformal restriction". This is a mathematical theory that was recently developed within the context of the Schramm-Loewner evolution which describes the "stochastic geometry" of fractal curves and other stochastic geometrical fractal objects in two-dimensional space. Observables elucidating the connection with the plateau transition include the so-called point-contact conductances (PCCs) between points on the boundary of the sample, described within the language of the Chalker-Coddington network model for the transition. We show that the disorder-averaged PCCs are characterized by a classical probability distribution for certain geometric objects in the plane (which we call pictures), occurring with positive statistical weights, that satisfy the crucial so-called restriction property with respect to changes in the shape of the sample with absorbing boundaries; physically, these are boundaries connected to ideal leads. At the transition point, these geometrical objects (pictures) become fractals. Upon combining this restriction property with the expected conformal invariance at the transition point, we employ the mathematical theory of "conformal restriction measures" to relate the disorder-averaged PCCs to correlation functions of (Virasoro) primary operators in a conformal field theory (of central charge c = 0). We show how this can be used to calculate these functions in a number of geometries with various boundary conditions. Since our results employ only the conformal restriction property, they are equally applicable to a number of other critical disordered electronic systems in two spatial dimensions, including for example the spin quantum Hall effect, the thermal metal phase in symmetry class D, and classical diffusion in two dimensions in a perpendicular magnetic field. For most of these systems, we also predict exact values of critical exponents related to the spatial behavior of various disorder-averaged PCCs.

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Critical exponent for the quantum Hall transition

Cited by 159 publications

References 25 publications

Topological quantum number and critical exponent from conductance fluctuations at the quantum Hall plateau transition

Topological quantum number and critical exponent from conductance fluctuations at the quantum Hall plateau transition

Weakly chiral networks and two-dimensional delocalized states in a weak magnetic field

Quantum Hall transitions: An exact theory based on conformal restriction

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