Magneto-roton theory of collective excitations in the fractional quantum Hall effect

Girvin, S. M.; MacDonald, Allan H.; Platzman, P. M.

doi:10.1103/physrevb.33.2481

Cited by 746 publications

(761 citation statements)

References 44 publications

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“…This progress was made possible by the non trivial algebra obeyed by its projected density operators [16]. For a smooth enough Berry curvature in the Brillouin zone (BZ), this algebra is nothing but the celebrated Girvin-MacDonald-Plazmann (GMP) algebra of the Fractional Quantum Hall effect [17]. This algebra has far reaching consequences: it is identical to the algebra of area-preserving diffeomorphisms, thereby providing for an explanation of the edge modes of an integer quantum Hall liquid as shape deformations of the liquid droplet.…”

Section: Introductionmentioning

confidence: 99%

D-algebra structure of topological insulators

2012

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In the quantum Hall effect, the density operators at different wave-vectors generally do not commute and give rise to the Girvin MacDonald Plazmann algebra with important consequences such as ground-state center of mass degeneracy at fractional filling fraction, and W1+∞ symmetry of the filled Landau levels. We show that the natural generalization of the GMP algebra to higher dimensional topological insulators involves the concept of a D-commutator. For insulators in even dimensional space, the D-commutator is isotropic and closes, and its structure factors are proportional to the D/2-Chern number. In odd dimensions, the algebra is not isotropic, contains the weak topological insulator index (layers of the topological insulator in one less dimension) and does not contain the Chern-Simons θ form. This algebraic structure paves the way towards the identification of fractional topological insulators through the counting of their excitations. The possible relation to D-dimensional volume preserving diffeomorphisms and parallel transport of extended objects is also discussed.

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Section: Introductionmentioning

confidence: 99%

D-algebra structure of topological insulators

2012

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“…For example, it is well known from the classic work of Girvin, MacDonald and Platzman [21] that the spectrum of collective modes with a finite length scale, such as the roton modes, is due to a combination of the magnetic algebra and electron-electron interaction effects.…”

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confidence: 99%

Non-commutative Chern–Simons for the quantum Hall system and duality

2002

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The quantum Hall system is known to have two mutually dual Chern-Simons descriptions, one associated with the hydrodynamics of the electron fluid, and another associated with the statistics. Recently, Susskind has made the claim that the hydrodynamic Chern-Simons theory should be considered to have a non-commutative gauge symmetry. The statistical Chern-Simons theory has a perturbative momentum expansion. In this paper, we study this perturbation theory and show that the effective action, although commutative at leading order, is non-commutative. This conclusion is arrived at through a careful study of the three-point function of ChernSimons gauge fields. The non-commutative gauge symmetry of this system is thus a quantum symmetry, which we show can only be fully realized only through the inclusion of all orders in perturbation theory. We discuss the duality between the two non-commutative descriptions.

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“…The low lying excited states of these quantum liquids are believed to consist of collective modes with a deep minimum in their dispersion relation, known as the magnetoroton minimum, at wavevectors comparable to l À1 f where l f is the magnetic length [6]. In recent years these excitations have been investigated experimentally by means of resonant Raman scattering techniques [7] and time-averaged phonon absorption spectroscopy [1].…”

Section: Physica B 256±258 (1998) 36±42mentioning

confidence: 99%

Angle-resolved ballistic phonon absorption spectroscopy in the lowest Landau level

Mellor

Zeitler

Devitt

et al. 1999

Physica B: Condensed Matter

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We report the results of a study of the absorption of non-equilibrium ballistic phonons by a two-dimensional electron system, in strong magnetic ®elds, by measuring its change in resistance. At the even denominator Landau level ®lling factors, m, of 1/2 and 3/2 the electron±phonon interaction is observed to be cut-o above a particular value of the component of the wavevector perpendicular to the low dimensional system. This cut-o is consistent with that due to the ®nite thickness of the 2DES. At fractional quantum Hall e ect minima (1/3, 2/5, 2/3) we observe that the cut-o to the interaction depends on the ®lling factor. At m 1/3, the energy gap determined from these experiments is consistent with the theoretical prediction of the magnetoroton energy gap. Ó

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Magneto-roton theory of collective excitations in the fractional quantum Hall effect

Cited by 746 publications

References 44 publications

D-algebra structure of topological insulators

D-algebra structure of topological insulators

Non-commutative Chern–Simons for the quantum Hall system and duality

Angle-resolved ballistic phonon absorption spectroscopy in the lowest Landau level

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