Kubo Hall conductivity on a finite cylinder and the integer quantum hall effect

Hajdu, János; Janssen, Martin; Viehweger, O.

doi:10.1007/bf01303893

Cited by 35 publications

(15 citation statements)

References 19 publications

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“…Therefore, in the present work we take a different starting point: we investigate the Hall conductivity in an appropriate version for the cylinder geometry (cf. [14]) and show that in this case also it has a topological meaning as it can be expressed in terms of a winding number. This winding number has an intuitive interpretation: it is the number of oriented windings of those orbits, in a diagram of energy eigenvalues versus the center of mass coordinate, X, that connect opposite edge regions.…”

Section: Introductionmentioning

confidence: 93%

“…In our numerical calculations these corrections are still visible for system sizes of order 100a. Equation (13) has been analyzed previously in the context of continuous electron systems on the surface of a cylinder [14,19,18]. We mention that the very fact of level anti-crossing leads to the presence of energy gaps (mini gaps) in the regime of edge states.…”

Section: Topology and The Quantum Hall Effectmentioning

confidence: 99%

“…The parameter ϑ ∈ [0, 1) generalizes the periodicity of the wavefunction to be periodic "up to a phase". ϑ/M y can, however, also be interpreted as a Bloch quantum number [14]. Indeed, by transforming the eigenvalue problem Hψ = Eψ according to the replacement…”

Section: Introductionmentioning

confidence: 99%

“…With y denoting the coordinate along the circumference of the cylinder, ϑ describes an Aharonov-Bohm flux (in units of the flux quantum) along the cylinder axis (cf. [14]).…”

Section: Introductionmentioning

confidence: 99%

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Lattice electrons on a cylinder surface in the presence of rational magnetic flux and disorder

et al. 1997

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We consider a disordered two-dimensional system of independent lattice electrons in a perpendicular magnetic field with rigid confinement in one direction and generalized periodic boundary conditions (GPBC) in the other direction. The objects investigated numerically are the orbits in the plane spanned by the energy eigenvalues and the corresponding center of mass coordinate in the confined direction, parameterized by the phase characterizing the GPBC. The Kubo Hall conductivity is expressed in terms of the winding numbers of these orbits. For vanishing disorder the spectrum of the system consists of Harper bands with energy levels corresponding to the edge states within the band gaps. Disorder leads to broadening of the bands. For sufficiently large systems localized states occur in the band tails. We find that within the mobility gaps of bulk states the Diophantine equation determines the value of the Hall conductivity as known for systems with torus geometry (PBCs in both directions). Within the spectral bands of extended states the Hall conductivity fluctuates strongly. For sufficiently large systems the generic behavior of localization-delocalization transitions characteristic for the quantum Hall effect are recovered.

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Section: Introductionmentioning

confidence: 93%

Section: Topology and The Quantum Hall Effectmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…With y denoting the coordinate along the circumference of the cylinder, ϑ describes an Aharonov-Bohm flux (in units of the flux quantum) along the cylinder axis (cf. [14]).…”

Section: Introductionmentioning

confidence: 99%

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Lattice electrons on a cylinder surface in the presence of rational magnetic flux and disorder

et al. 1997

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“…7 Although Stfeda's formula is derived for a continuous Hamiltonian, it is easily seen to hold for the case in hand. Indeed, as pointed out by Hadju etal in [19], Stfeda implicitly assumes that the trace of the projection is uniformly bounded when he interchanges two limits, and whilst this is not a problem for either a confined system (as considered in [19]) or for a discrete system as we are considering here, it requires further justification in the case of particles moving in R 2 without a confining potential.…”

Section: § 42 Conductivity For the Discrete Modelmentioning

confidence: 99%

A Discrete Model of the Integer Quantum Hall Effect

McCann

Carey

1996

Publ. Res. Inst. Math. Sci.

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A discrete model of the integer quantum Hall effect is analysed via its associated C*-algebra. The relationship with the usual continuous models is established by viewing the observable algebras of each as both twisted group C*-algebras and twisted cross products. A Fredholm module for the discrete model is presented, and its Chern character is calculated. § 1. IntroductionThe discovery of the integer quantum Hall effect has prompted a wealth of theoretical speculation about the origin of the spectacular accuracy with which the Hall conductance is quantized. This paper presents a simple lattice model of the quantum Hall effect that generates much of the information arising from more complex models. This lattice model of the quantum Hall effect is often used as the discrete analogue of the Landau Hamiltonian in the physics literature, and the analysis of the model often requires restricting to rational values of the magnetic flux. It is here extended and recast to fit into the C*-algebraic framework, a development that allows (in § 3) the Hall conductance to be calculated for all real values of flux. The analysis of the expression for the conductance makes its stability with respect to small changes in magnetic field evident, for it is found to be the Chern number associated with the Fermi projection (when the latter lies in a gap of the spectrum of the discrete Hamiltonian). We display the equivalence with the formula found for rational flux in the physics literature by using an explicit representation of the algebra of observables.The Hall effect is often modelled by considering electrons moving on a plane under the influence of a perpendicular magnetic field and a periodic potential. We show in § 2 that for both this model and the discrete model mentioned above the

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Linear response, persistent currents and related problems

et al. 1993

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Abstract. Based on a general linear response approach we provide a systematic and unified survey of existing theories on persistent currents. The central notions in this context are equilibrium and dynamic persistent currents which are analyzed with respect to their similarities and differences in the canonical and grand canonical ensemble. We present criteria which relate the existence of persistent currents to the equipartition law and ergodicity for current correlators. We find that in additive Fermion systems at low temperatures both kinds of persistent currents coincide in the canonical ensemble whereas they differ in the grand canonical ensemble. Comparing different works on averaged persistent currents in diffusive mesoscopic rings within our framework and discussing several methods of calculating canonical currents with the help of grand canonical ensembles, we clarify some misunderstandings which have arisen in methodologically different approaches to the phenomenon of persistent currents. Finally, we relate the presence of dynamic persistent currents to the Hall conductivity on a finite cylinder and the center coordinate Kubo formula for the Hall conductivity.

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Kubo Hall conductivity on a finite cylinder and the integer quantum hall effect

Cited by 35 publications

References 19 publications

Lattice electrons on a cylinder surface in the presence of rational magnetic flux and disorder

Lattice electrons on a cylinder surface in the presence of rational magnetic flux and disorder

A Discrete Model of the Integer Quantum Hall Effect

Linear response, persistent currents and related problems

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