Chiral partition functions of quantum Hall droplets

Cappelli, Andrea; Viola, Giovanni; Zemba, Guillermo R.

doi:10.1016/j.aop.2009.10.007

Cited by 22 publications

(69 citation statements)

References 60 publications

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“…Modular invariant partition functions were obtained in [69,70,71,72] for the Abelian hierarchical and non-Abelian Read-Rezayi states; in section 3.2, we provide the expressions for the other non-Abelian states in the second Landau level for 2 < ν < 3 (these results was obtained in [73]). These are the non-Abelian spin-singlet states (NASS), introduced in [15] and further developed in [58], the charge-conjugates of Read-Rezayi's states (RR) [74], the Bonderson-Slingerland (BS) hierarchy built over the Pfaffian state [75], and Wen's SU (n) non-Abelian fluids (NAF) [76], the SU (2) case in particular.…”

Section: Chapter 3 Modular Invariant Partition Function In Qhementioning

confidence: 99%

“…We start with some general considerations on the partition function for QH edges states and use two well understood cases as examples, the Laughlin [69] and the ReadRezayi states [67,72]. …”

Section: Chapter 3 Modular Invariant Partition Function In Qhementioning

confidence: 99%

“…The torus geometry enjoys the symmetry under modular transformations acting on the two periodic coordinates and basically exchanging the space and time periods. The partition functions should be invariant under this geometrical symmetry that actually implies several physical conditions on the spectrum of the theory [69,72].…”

Section: Partition Functions For Hall Fluids 311 Annulus Geometrymentioning

confidence: 99%

“…The final step is to find the modular transformations of θ ℓ a , that is given by [72] (see Appendix A):…”

Section: Read-rezayi Statesmentioning

confidence: 99%

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Partition functions of non-Abelian quantum Hall states

Cappelli¹,

Viola²

2011

J. Phys. A: Math. Theor.

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OverviewThe fractional quantum Hall effect is a collective quantum phenomenon of two-dimensional electrons placed in a strong perpendicular magnetic field (B ∼ 1 − 10 Tesla) and at very low temperature (T ∼ 50 − 500 mK). It manifests itself in the measure of the transverse (σ xy ) and longitudinal (σ xx ) conductances as functions of the magnetic field. The Hall conductance σ xy shows the characteristic step-like behavior: at each step (plateau), it is quantized in rational multiples of the quantum unit of conductance e 2 /h: σ xy = ν e 2 h ; furthermore, at the plateaus centers the longitudinal resistance vanishes. These features are independent on the microscopic details of the samples; in particular, the quantization of σ xy is very accurate, with experimental errors of the order 10 −8 Ω. The rational number ν is the filling fraction of occupied one-particle states at the given value of magnetic field.The fractional Hall effect is characterized by a non-perturbative gap due to the Coulomb interaction among electrons in strong fields. The microscopic theory is impracticable, besides numerical analysis; thus, effective theories, effective field theories in particular, are employed to study these systems. The first step was made by Laughlin, who proposed the trial ground state wave function for fillings ν = , · · · describing the main physical properties of the Hall effect. One general feature is that the electrons form a fluid with gapped excitations in the bulk, the so-called incompressible fluid. In a finite sample, the incompressible fluid gives rise to edge excitations, that are gapless and responsible for the conduction properties. Therefore the low-energy effective field theory should describe these edge degrees of freedom.The Laughlin theory revealed that the excitations are "anyons", i.e. quasiparticles with fractional values of charge and exchange statistics. The latter is a possibility in bidimensional quantum systems, where the statistics of particles is described by the braid group instead of the permutation group of higher-dimensional systems. Each filling fraction corresponds to a different state of matter that is characterized by specific fractional values of charge and statistics. The braiding of quasiparticles can also occur for multiplets of degenerate excitations, by means of multi-dimensional unitary transformations; in this case, called non-Abelian fractional statistics, two different braidings do not commute. The corresponding non-Abelian quasiparticles are degenerate for fixed positions and given quantum numbers.Non-Abelian anyons are candidate for implementing topological quantum computations according to the proposal by Kitaev and others. Since Anyons are non-perturbative collective excitations, they are less affected by decoherence due to local disturbances. III IVExperimental observation of non-Abelian statistics is a present challenge.The topological fluids, such as the quantum Hall fluid can be described by the ChernSimon effective field theory in the low-energy long-range limit. Th...

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Section: Chapter 3 Modular Invariant Partition Function In Qhementioning

confidence: 99%

Section: Chapter 3 Modular Invariant Partition Function In Qhementioning

confidence: 99%

Section: Partition Functions For Hall Fluids 311 Annulus Geometrymentioning

confidence: 99%

“…The final step is to find the modular transformations of θ ℓ a , that is given by [72] (see Appendix A):…”

Section: Read-rezayi Statesmentioning

confidence: 99%

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Partition functions of non-Abelian quantum Hall states

Cappelli¹,

Viola²

2011

J. Phys. A: Math. Theor.

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“…In this paper, we reserve µ E as a real number and denote iµ E when we consider an imaginary entanglement chemical potential. An imaginary chemical potential has been used in quantum Hall systems, for instance 22 , and QCD, for instance 23 . One advantage of using the imaginary potential is that in the case where the conserved charge corresponds to angular momenta, which we will consider in the next subsection, a real chemical potential could lead to unwanted divergences in the partition function which come from excitations moving faster than the speed of light at fixed angular potential in a non-compact space.…”

Section: Grand Canonical Entanglement Entropies and Flux Operatorsmentioning

confidence: 99%

Charged topological entanglement entropy

et al. 2016

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A charged entanglement entropy is a new measure which probes quantum entanglement between different charge sectors. We study symmetry protected topological (SPT) phases in 2+1 dimensional space-time by using this charged entanglement entropy. SPT phases are short range entangled states without topological order and hence cannot be detected by the topological entanglement entropy. We demonstrate that the universal part of the charged entanglement entropy is non-zero for nontrivial SPT phases and therefore it is a useful measure to detect short range entangled topological phases. We also discuss that the classification of SPT phases based on the charged topological entanglement entropy is related to that of the braiding statistics of quasiparticles.

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Hilbert Space Decomposition for Coulomb Blockade in Fabry–Pérot Interferometers

Georgiev

2013

Springer Proceedings in Mathematics &Amp; Statistics

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We show how to construct the thermodynamic grand potential of a droplet of incompressible fractional quantum Hall liquid, formed inside of an electronic Fabry-Pérot interferometer, in terms of the conformal field theory disk partition function for the edge states in presence of Aharonov-Bohm flux. To this end we analyze in detail the algebraic structure of the edge states' Hilbert space and identify the effect of the variation of the flux. This allows us to compute, in the linear response approximation, all thermodynamic properties of the conductance in the regime when the Coulomb blockade is softly lifted by the change of the magnetic flux due to the weak coupling between the droplet and the two quantum point contacts.

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Chiral partition functions of quantum Hall droplets

Cited by 22 publications

References 60 publications

Partition functions of non-Abelian quantum Hall states

Partition functions of non-Abelian quantum Hall states

Charged topological entanglement entropy

Hilbert Space Decomposition for Coulomb Blockade in Fabry–Pérot Interferometers

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