A Topological Look at the Quantum Hall Effect

Avron, J. E.; Osadchy, D.; Seiler, R.

doi:10.1063/1.1611351

Cited by 189 publications

(167 citation statements)

References 18 publications

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“…The motion of the bosons on the geometrically frustrated Shastry-Sutherland lattice is therefore akin to orbital confinement in two-dimensional quantum Hall systems by a (statistical) magnetic field (24,25). Plateaus with a finite Hall conductance arise when the chemical potential ( ϭ g c B H) resides in a gap of the energy spectrum of fermions with average density ϭ m z ϩ 1/2 ϭ a 2 H s /4⌽ 0 (a 2 is the area of the unit cell).…”

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confidence: 99%

Fractalization drives crystalline states in a frustrated spin system

Sebastian

Harrison

Sengupta

et al. 2008

Proc. Natl. Acad. Sci. U.S.A.

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We measure a sequence of quantum Hall-like plateaux at 1/q: 9 >= q >= 2 and p/q = 2/9 fractions in the magnetisation with increasing magnetic field in the geometrically frustrated spin system SrCu2(BO3)2. We find that the entire observed sequence of plateaux is reproduced by solving the Hofstadter problem on the system lattice when short-range repulsive interactions are included, thus providing a sterling demonstration of bosons confined by a magnetic and lattice potential mimicking fermions in the extreme quantum limit.Comment: 14 pages, 3 figure

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confidence: 99%

Fractalization drives crystalline states in a frustrated spin system

Sebastian

Harrison

Sengupta

et al. 2008

Proc. Natl. Acad. Sci. U.S.A.

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“…17 In addition, it has an elegant geometric interpretation as the integral of the curvature of the quantum state in the parameter space spanned by two angular parameters ͑twisted boundary conditions͒-the first Chern class of a U͑1͒ principal fiber bundle on the torus. [24][25][26] Chern numbers are topologically invariant under small perturbations of Hamiltonian, such as weak disorder, which allows the mobility gap to open for a finite range of magnetic field, leading to the plateau structure of the IQHE as long as the Fermi level lies in the mobility gap.…”

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Mobility gap in fractional quantum Hall liquids: Effects of disorder and layer thickness

et al. 2005

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We study the behavior of two-dimensional electron gas in the fractional quantum Hall regime in the presence of finite layer thickness and correlated disordered potential. Generalizing the Chern number calculation to many-body systems, we determine the mobility gaps of fractional quantum Hall states based on the distribution of Chern numbers in a microscopic model. We find excellent agreement between experimentally measured activation gaps and our calculated mobility gaps, when combining the effects of both disordered potential and layer thickness. We clarify the difference between mobility gap and spectral gap of fractional quantum Hall states and explain the disorder-driven collapse of the gap and the subsequent transitions from the fractional quantum Hall states to the insulator.

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“…7]. Unlike the quantum Hall effect [5], it occurs in the absence of a magnetic field and has been predicted to affect neutral Helium atoms [6].Conductance quantization has recently been observed in ultracold fermionic gases [7]. Atomic gases allow for a clean observation in a simple setup involving two reservoirs connected by a constriction within which an attractive gate potential E G < 0 is varied (see Fig.…”

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Quantized conductance through the quantum evaporation of bosonic atoms

2016

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We analyze theoretically the quantization of conductance occurring with cold bosonic atoms trapped in two reservoirs connected by a constriction with an attractive gate potential. We focus on temperatures slightly above the condensation threshold in the reservoirs. We show that a conductance step occurs, coinciding with the appearance of a condensate in the constriction. Conductance relies on a collective process involving the quantum condensation of an atom into an elementary excitation and the subsequent quantum evaporation of an atom, in contrast with ballistic fermion transport. The value of the bosonic conductance plateau is strongly enhanced compared to fermions and explicitly depends on temperature. We highlight the role of the repulsive interactions between the bosons in preventing them from collapsing into the constriction.PACS numbers: 05.60. Gg,05.30.Jp, In mesoscopic systems, where the motion of quantum particles occurs over distances of the order of their coherence length, transport phenomena exhibit quantum signatures [1]. The quantization of conductance [2] is a hallmark among these effects. It reflects the discrete nature of the transport channels inside a strongly constricted geometry, and it occurs if the spread in energies of the incident particle distribution is smaller than the energy separation of these channels. It was first observed in electronic transport through a quantum point contact [3] as a series of plateaux in the conductance when the distance between the gate electrodes was increased. In this fermionic case, the conductance quantum G K = e 2 /h involves fundamental constants only, which makes it relevant for metrology [4, chap. 7]. Unlike the quantum Hall effect [5], it occurs in the absence of a magnetic field and has been predicted to affect neutral Helium atoms [6].Conductance quantization has recently been observed in ultracold fermionic gases [7]. Atomic gases allow for a clean observation in a simple setup involving two reservoirs connected by a constriction within which an attractive gate potential E G < 0 is varied (see Fig. 1). Experiments on ultracold fermions aim at simulating electronic systems using neutral particles [7][8][9][10]. In the fermionic experiment of Ref. [7], conductance quantization has been observed at temperatures much lower than both the Fermi temperature T F and the confinement energy of the constriction, in analogy with the original results on electronic transport [3] where only particles near the Fermi surface take part in transport phenomena. This raises the question of whether conductance quantization also affects bosons. Previous observations in an optical setup [11,12] and predictions with cold matter waves [13] have focused on systems where all particles have the same incident energy, mimicking fermionic transport at the Fermi energy. To our knowledge, the specific role of bosonic statistics in quantized conductance situations has not yet been investigated. Cold atom setups allow for theTwo reservoirs (L, R) can exchange particles through a s...

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A Topological Look at the Quantum Hall Effect

Abstract: The amazingly precise quantization of Hall conductance in a two-dimensional electron gas can be understood in terms of a topological invariant known as the Chern number.

Cited by 189 publications

References 18 publications

Fractalization drives crystalline states in a frustrated spin system

Fractalization drives crystalline states in a frustrated spin system

Mobility gap in fractional quantum Hall liquids: Effects of disorder and layer thickness

Quantized conductance through the quantum evaporation of bosonic atoms

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