Universal Singularities in the Integral Quantum Hall Effect

Pruisken, A. M. M.

doi:10.1103/physrevlett.61.1297

Cited by 343 publications

(310 citation statements)

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“…Thouless showed that localized states have zero Chern numbers 12 , which is easy to understand, because localized states are rather insensitive to changes in the boundary conditions used to calculate the Chern number 9 . This is verified by numerical calculations 10,11 , which find that non-zero Chern numbers appear near the centre of the Landau level, with a distribution in perfect agreement with the scaling theory of the integer quantum Hall effect [13][14][15] (IQHE).…”

supporting

confidence: 55%

Spectral weight transfer in the integer quantum Hall effect and its consequences

Zhou

Berciu

2007

Nature Phys

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The energy spectrum of a two-dimensional electron gas placed in a transversal magnetic field B consists of quantized Landau levels. In the absence of disorder, the degeneracy of each Landau level is N = B A/φ 0 , where A is the area of the sample and φ 0 = h/e is the magnetic flux quantum. With disorder, localized states appear at the top and bottom of the broadened Landau level, whereas states in the centre of the Landau level (the critical region) remain delocalized. This single-electron theory adequately explains most aspects of the integer quantum Hall effect 1 . One unnoticed issue is the location of the new states that appear in the Landau level with increasing B. Here, we show that they appear predominantly inside the critical region. This situation leads to a 'spectral ordering' of the localized states, which explains the stripes observed in measurements of the local inverse compressibility 2,3 , of two-terminal conductance 4 and of Hall and longitudinal resistances 5 without the need to invoke interactions as done in previous work 6-8 .The spectrum and eigenstates of a disorder-broadened Landau level can be studied with the well-established approach of diagonalizing the single-electron hamiltonianwhere V (r) is the disorder. We choose A(r) = (0, Bx) and apply periodic boundary conditions (PBCs) to a system of area A = L × L. Owing to the PBCs, the degeneracy of each Landau level, N = BL 2 /φ 0 , is an integer. Usually, in theoretical studies the magnetic field is kept fixed and the electron density n e or, equivalently, the filling factor ν = n e A/N, is swept by adjusting the Fermi energy E F .As the experiments mentioned above investigate the behaviour of various quantities in the (n e , B) plane, we need to understand how the spectrum changes when B is also tuned. Given the constraint that an integer number of fluxes must penetrate the sample, B can only change in discrete steps of φ 0 /L 2 . Therefore, we ask the following question: how do single-electron wavefunctions evolve when one more magnetic flux is inserted?Let |i, N , 1 ≤ i ≤ N be the eigenstates of a spin-polarized Landau level corresponding to a given disorder V (r) and a magnetic field B = N φ 0 /L 2 . The states are ordered by their energies E 1 < E 2 < ··· < E N (accidental degeneracies can be lifted with minute changes in V (r)).To see how the wavefunctions evolve when B increases, we calculate their disorder-averaged overlaps:where 1 ≤ i ≤ N and 1 ≤ j ≤ N + 1 label two eigenstates of the hamiltonian (1) with the same disorder potential but different magnetic fields. The overline indicates a disorder average. In the results presented here we typically average over 1,000 disorder realizations, and show results only for the spin-polarized lowest Landau level. Similar results are expected in higher Landau levels. The disorder potential V (r) is modelled as a sum of many shortrange, randomly placed gaussian scatterers. We show four sets of data, for L = 250, 400, 500, 750 nm, N = 50, 128, 200, 550 and therefore B = 1.654 T for the first t...

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supporting

confidence: 55%

Spectral weight transfer in the integer quantum Hall effect and its consequences

Zhou

Berciu

2007

Nature Phys

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“…Here, the error is less than 10% in a Gaussian model of the density of states. The theory predicts a complex conductivity following a scaling function s xx ͑f, dn͒ g xx ͓L f ͞j͑dn͔͒ [13], where L f~f 2z is the frequency dependent dynamic length. Since Re͑s xx ͒ depends monotonically on dn we can invert Res xx ͑y͒ and replace the argument y L f ͞j͑dn͒ in Ims xx ͑y͒ with some function g 21 ͑Res xx ͒.…”

Section: High Frequency Conductivity In the Quantum Hall Regimementioning

confidence: 99%

High Frequency Conductivity in the Quantum Hall Regime

2001

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We have measured the complex conductivity s xx of a two-dimensional electron system in the quantum Hall regime up to frequencies of 6 GHz at electron temperatures below 100 mK. Using both its imaginary and real part we show that s xx can be scaled to a single function for different frequencies and several transitions between plateaus in the quantum Hall effect. Additionally, the conductivity in the variable-range hopping regime is used for a direct evaluation of the localization length j. Even for large filling factor distances dn from the critical point we find j~dn 2g with a scaling exponent g 2.3. DOI: 10.1103/PhysRevLett.86.5124 PACS numbers: 73.43.-f, 71.23.An, 71.30.+h, 72.20.My It is widely accepted that the understanding of the integer quantum Hall effect (QHE) is closely related to a disorder driven localization-delocalization transition occurring in two-dimensional electron systems (2DES) in high magnetic fields [1]. Many experimental and theoretical works approve the interpretation of the transition between adjacent QHE plateaus as a quantum critical phase transition. It is governed by a diverging localization length j~jE 2 E c j 2g which scales with the distance of the energy E from the critical energy E c in the center of a Landau band. The exponent g ഠ 2.3 is believed to be a universal quantity independent of disorder. For finite systems with effective size L eff theory predicts that the conductivities s ab follow scaling functions s ab f ab ͓L eff ͞j͑E͔͒ resulting in a finite width DE~L 21͞g eff of the transition region. The effective system size L eff is determined by the physical sample size, the electron temperature T , or the frequency f.The most common test of scaling uses an analysis of the temperature or frequency dependence of the conductivity peak width in the QHE plateau transition. However, lacking an exact expression for L eff ͑T , f͒, this method does not allow direct access to the scaling behavior of the localization length.An alternative approach to scaling was proposed by Polyakov and Shklovskii [2]. Using the fact that the conductivity s xx in the QHE plateaus at low temperatures is dominated by variable-range hopping (VRH) [3] a direct access to the localization length j can be gained from analyzing the dependence of s xx on temperature, current, and frequency. Experimentally mainly the temperature dependence of s xx in the VRH regime was investigated [4]. However, due to an unknown theoretical prefactor, j could only be estimated from these experiments.In contrast, the frequency driven variable-range hopping conductivity s xx ͑f͒ [in the limit s xx ͑f͒ ¿ s xx ͑0͒] is given by [2] Res xx ͑v͒ 2p 3 ee 0 jv ,linearly depending on both frequency f v͞2p and localization length j with no unknown prefactors.Here we report on measurements of the complex conductivity s xx up to frequencies f 6 GHz at low temperatures down to below 100 mK. We will show that Im͑s xx ͒ can be scaled with a single-parameter function to Re͑s xx ͒, independent of temperature, frequency, and filling factor. ...

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“…The integrally quantized Hall plateaus (IQHP) are observed when the Fermi level lies in localized states, with the value of the Hall conductance, σ xy = ne 2 /h, related to the number of occupied extended states(n). Many previous studies 3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 have been focused on so-called plateau transitions. The issue there is how the Hall conductance jumps from one quantized value to another when the Fermi level crosses an extended state.…”

Section: Introductionmentioning

confidence: 99%

Possible existence of a band of extended states induced by inter-Landau-band mixing in a quantum Hall system

Xiong

Wang

Niu

et al. 2006

J. Phys.: Condens. Matter

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The mixing of states with opposite chirality in quantum Hall system is shown to have delocalization effect. It is possible that extended states may form bands because of this mixing, as is shown through a numerical calculation on a two-channel network model. Based on this result, a new phase diagram with a narrow metallic phase separating two adjacent QH phases and/or separating a QH phase from the insulating phase is proposed. The data from recent non-scaling experiments is reanalyzed and shown that they seem to be consistent with the new phase diagram. However, due to finite-size effects, further study on large system size is still needed to conclude whether there are extended state bands in thermodynamic limit.

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Universal Singularities in the Integral Quantum Hall Effect

Cited by 343 publications

References 10 publications

Spectral weight transfer in the integer quantum Hall effect and its consequences

Spectral weight transfer in the integer quantum Hall effect and its consequences

High Frequency Conductivity in the Quantum Hall Regime

Possible existence of a band of extended states induced by inter-Landau-band mixing in a quantum Hall system

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