Universality in quantum Hall systems

Fröhlich, Jürg; Kerler, Thomas

doi:10.1016/0550-3213(91)90360-a

Cited by 235 publications

(291 citation statements)

References 63 publications

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“…Thus p = 1 corresponds to the Laughlin series. In particular, also for p = 1, this is the form of the action that follows from hydrodynamic arguments [6,11,12,13,14]. It is straightforward to show [8] that this theory leads to the correct value of the Hall conductivity…”

Section: The Non-commutative Effective Action For the Statisticalmentioning

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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Section: The Non-commutative Effective Action For the Statisticalmentioning

confidence: 99%

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

2002

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“…Using this point of view, we have developed a systematic and quantitative theory for topological orders with gappable edge for (2+1)-dimensional [(2+1)D] interacting boson and fermion systems [9,12,27]. Also, for (2+1)D topological orders with only Abelian statistics, we find that we can use integer K-matrices to describe them [29][30][31][32].…”

Section: A Short-and Long-range Entangled Statesmentioning

confidence: 99%

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory

2014

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Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have T 2 = 1 time-reversal Z T 2 and fermion-number-parity Z f 2 symmetries described by a full symmetry group Z T 2 × Z f 2 ). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the K-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group Z 2 × Z f 2 . Those 2D fermionic SPT phases all have central-charge c = 1 gapless edge excitations, if the symmetry is not broken.

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“…where q can assume any integer value [9,10], and may be thought of as the effective number of Pauli-Villars regulators, namely, the number of regulators with positive mass minus the number of negative mass ones. Adding a gauge-fixing term λ 2 (∂ · b) 2 , the b -dependent part of the path integral (in momentum-space) reads:…”

mentioning

confidence: 99%

“…This sort of ambiguity, which manifests itself in the bosonization rules, already exists in the fermionic description. It is due to the regularization dependence of the induced Chern-Simons term [9].…”

mentioning

confidence: 99%

On bosonization in 3 dimensions

Barci¹,

Fosco²,

Oxman³

1996

Physics Letters B

101

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A recently proposed path-integral bosonization scheme for massive fermions in 3 dimensions is extended by keeping the full momentum-dependence of the one-loop vacuum polarization tensor. This makes it possible to discuss both the massive and massless fermion cases on an equal footing, and moreover the results it yields for massless fermions are consistent with the ones of another, seemingly different, canonical quantization approach to the problem of bosonization for a massless fermionic field in 3 dimensions.

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Universality in quantum Hall systems

Cited by 235 publications

References 63 publications

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

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

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory

On bosonization in 3 dimensions

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