An experimental study on Γ(2) modular symmetry in the quantum Hall system with a small spin splitting

Huang, Chi‐Feng; Chang, Yen-Jen; Cheng, Hao-Tien; Yang, Zu-Po; Yeh, H. D.; Hsu, Chuang‐Han; Liang, Chi‐Te; Hang, Da-Ren; Lin, H. H.

doi:10.1088/0953-8984/19/2/026205

Cited by 11 publications

(27 citation statements)

References 44 publications

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“…3 Spin effects can also modify the precise position of the critical points [17,32]. 4 Because this argument only relies on using duality to relate the conductivity produced by a vortex with that produced by a quasi-particle -as opposed to trying to explicitly compute either result separately, as Figure 7: A plot of some of the flow lines (for decreasing temperature) for the conductivities that are dictated by Γ 0 (2) invariance.…”

Section: Some Relevant Experimentsmentioning

confidence: 99%

AdS/QHE: towards a holographic description of quantum Hall experiments

et al. 2011

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Transitions among quantum Hall plateaux share a suite of remarkable experimental features, such as semi-circle laws and duality relations, whose accuracy and robustness are difficult to explain directly in terms of the detailed dynamics of the microscopic electrons. They would naturally follow if the low-energy transport properties were governed by an emergent discrete duality group relating the different plateaux, but no explicit examples of interacting systems having such a group are known. Recent progress using the AdS/CFT correspondence has identified examples with similar duality groups, but without the DC ohmic conductivity characteristic of quantum Hall experiments. We use this to propose a simple holographic model for low-energy quantum Hall systems, with a nonzero DC conductivity that automatically exhibits all of the observed consequences of duality, including the existence of the plateaux and the semi-circle transitions between them. The model can be regarded as a strongly coupled analog of the old 'composite boson' picture of quantum Hall systems. Nonuniversal features of the model can be used to test whether it describes actual materials, and we comment on some of these in our proposed model.

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Section: Some Relevant Experimentsmentioning

confidence: 99%

AdS/QHE: towards a holographic description of quantum Hall experiments

et al. 2011

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“…For the particular case of the ν : 1 → 0 transition any re-scaling of ρ xx still gives a semi-circle in the σ-plane, but this is not true for other transitions. The observation that the critical point is not at σ = (3+i)/2 for the ν : 2 → 1 transition in [109], as shown in Fig. 9, cannot be explained by a re-scaling of the Ohmic resistivity and therefore is 15 Because of the spin degeneracy the transformations are σ → σ + 2 and σ → σ 1−σ , i.e.…”

Section: Duality and The Quantum Hall Ef Fectmentioning

confidence: 94%

Modular Symmetry and Fractional Charges in N = 2 Supersymmetric Yang-Mills and the Quantum Hall Effect

Dolan¹

2007

SIGMA

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Abstract. The parallel rôles of modular symmetry in N = 2 supersymmetric Yang-Mills and in the quantum Hall effect are reviewed. In supersymmetric Yang-Mills theories modular symmetry emerges as a version of Dirac's electric -magnetic duality. It has significant consequences for the vacuum structure of these theories, leading to a fractal vacuum which has an infinite hierarchy of related phases. In the case of N = 2 supersymmetric Yang-Mills in 3 + 1 dimensions, scaling functions can be defined which are modular forms of a subgroup of the full modular group and which interpolate between vacua. Infra-red fixed points at strong coupling correspond to θ-vacua with θ a rational number that, in the case of pure SUSY Yang-Mills, has odd denominator. There is a mass gap for electrically charged particles which can carry fractional electric charge. A similar structure applies to the 2 + 1 dimensional quantum Hall effect where the hierarchy of Hall plateaux can be understood in terms of an action of the modular group and the stability of Hall plateaux is due to the fact that odd denominator Hall conductivities are attractive infra-red fixed points. There is a mass gap for electrically charged excitations which, in the case of the fractional quantum Hall effect, carry fractional electric charge.

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“…7 where the flow generated by Φ 0.93 has been superimposed on the data from Ref. 20. Notice that there is a cluster of data that is almost stationary, and therefore should be very close to an RG fixed point.…”

Section: A Spin Split Landau Levelmentioning

confidence: 95%

“…20. They found a critical point at ρ ⊗ = (7.75, 3.54) ± 0.05 kΩ ≈ (0.300, 0.137) ± 0.002 [h/e 2 ], which corresponds to a ρ = −1.76.…”

Section: A Spin Split Landau Levelmentioning

confidence: 99%

“…Using the same sample as in the experiment described above 19 , Huang et al 20 have studied the temperature scaling in more detail. The semi-circular geometry of the flows strongly suggests that the flow is following the separatrix connecting the plateaux at σ ⊕ = 1 and σ ⊕ = 2, if we allow for a systematic error that shifts σ D up by 0.17 e 2 /h, in which case a real value of the parameter a will suffice.…”

Section: A Spin Split Landau Levelmentioning

confidence: 99%

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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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An experimental study on Γ(2) modular symmetry in the quantum Hall system with a small spin splitting

Cited by 11 publications

References 44 publications

AdS/QHE: towards a holographic description of quantum Hall experiments

AdS/QHE: towards a holographic description of quantum Hall experiments

Modular Symmetry and Fractional Charges in N = 2 Supersymmetric Yang-Mills and the Quantum Hall Effect

Renormalization-group potential for quantum Hall effects

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