On localization in the theory of the quantized hall effect: A two-dimensional realization of the θ-vacuum

Pruisken, A. M. M.

doi:10.1016/0550-3213(84)90101-9

Cited by 269 publications

(258 citation statements)

References 13 publications

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“…The existence of such delocalized quantum states can be demonstrated 44 with mathematical rigor using the theory of non-commutative Chern number, 64 while numerically, it has been demonstrated using recursive Green's function and transfer matrix calculations, 51,74,99,100 level statistics analysis, 44,67,101 simulations of the edge currents and computations of the edge conductance. [102][103][104] Near the transitions, the field-theoretic arguments developed by Pruisken and collaborators [105][106][107][108] for IQHE predict the T-driven flowdiagram shown in Fig. 1(b), which was observed and confirmed in IQHE by experiment.…”

Section: Quantum Criticality In the Spin-up Sector Of The Kane-mmentioning

confidence: 81%

Quantum criticality at the Chern-to-normal insulator transition

Xue

Prodan

2013

Phys. Rev. B

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Using the non-commutative Kubo formula for aperiodic solids and a recently developed numerical implementation, we study the conductivity σ and resistivity ρ tensors as functions of Fermi level E F and temperature T, for models of strongly disordered Chern insulators. The formalism enabled us to converge the transport coefficients at temperatures low enough to enter the quantum critical regime at the Chern-to-trivial insulator transition. We find that the ρ xx -curves at different temperatures intersect each other at one single critical point, and that they obey a single-parameter scaling law with an exponent close to the universally accepted value for the unitary symmetry class. However, when compared with the established experimental facts on the plateau-insulator transition in the Integer Quantum Hall Effect, we find a universal critical conductance σ c xx twice as large, an ellipse rather than a semi-circle law, and absence of the Quantized Hall Insulator phase.

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Section: Quantum Criticality In the Spin-up Sector Of The Kane-mmentioning

confidence: 81%

Quantum criticality at the Chern-to-normal insulator transition

Xue

Prodan

2013

Phys. Rev. B

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“…Such models have arisen in various contexts: the integer quantum Hall (IQH) effect [1][2][3][4][5][6][7][8], and recent generalizations [9][10][11][12][13][14][15][16][17][18][19][20][21][22], as well as other problems of fermions with quenched disorder [23][24][25][26][27][28]; percolation, polymers, and other statistical mechanics problems (Refs. [29,13] and this paper); and strings in anti-de Sitter space (reviewed in Ref.…”

Section: Introductionmentioning

confidence: 99%

Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions

2001

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We study two-dimensional nonlinear sigma models in which the target spaces are the coset supermanifolds U(n + m|n)/[U(1)×U(n + m − 1|n)] ∼ = CP n+m−1|n (projective superspaces) and OSp(2n + m|2n)/OSp(2n + m − 1|2n) ∼ = S 2n+m−1|2n (superspheres), n, m integers, −2 ≤ m ≤ 2; these quantum field theories live in Hilbert spaces with indefinite inner products. These theories possess non-trivial conformally-invariant renormalization-group fixed points, or in some cases, lines of fixed points. Some of the conformal fixed-point theories can also be obtained within LandauGinzburg theories. We obtain the complete spectra (with multiplicities) of exact conformal weights of states (or corresponding local operators) in the isolated fixed-point conformal field theories, and at one special point on each of the lines of fixed points. Although the conformal weights are rational, the conformal field theories are not, and (with one exception) do not contain the affine versions of their superalgebras in their chiral algebras. The method involves lattice models that represent the strong-coupling region, which can be mapped to loop models, and then to a Coulomb gas with modified boundary conditions. The results apply to percolation, dilute and dense polymers, and other statistical mechanics models, and also to the spin quantum Hall transition in noninteracting fermions with quenched disorder.

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“…This was motivated by a sigmamodel of localization, which was extensively elaborated by Pruisken et al 38,39 , but the target space geometry does not appear to be rich enough to include the FQHE.…”

Section: Comparison With Other Workmentioning

confidence: 99%

Renormalization-group potential for quantum Hall effects

Nissinen

Lütken

2012

Phys. Rev. B

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The phenomenological analysis of fully spin-polarized quantum Hall systems, based on holomorphic modular symmetries of the renormalization group (RG) flow, is generalized to more complicated situations where the spin or other "flavors" of charge carriers are relevant, and where the symmetry is different. We make the simplest possible ansatz for a family of RG potentials that can interpolate between these symmetries. It is parametrized by a single number a and we show that this suffices to account for almost all scaling data obtained to date. The potential is always symmetric under the main congruence group at level two, and when a takes certain values this symmetry is enhanced to one of the maximal subgroups of the modular group. We compute the covariant RG β-function, which is a holomorphic vector field derived from the potential, and compare the geometry of this gradient flow with available temperature driven scaling data. The value of a is determined from experiment by finding the location of a quantum critical point, i.e., an unstable zero of the β-function given by a saddle point of the RG potential. The data are consistent with a ∈ R, which together with the symmetry leads to a generalized semi-circle law.

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On localization in the theory of the quantized hall effect: A two-dimensional realization of the θ-vacuum

Cited by 269 publications

References 13 publications

Quantum criticality at the Chern-to-normal insulator transition

Quantum criticality at the Chern-to-normal insulator transition

Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions

Renormalization-group potential for quantum Hall effects

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