Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States

Li, Hui; Haldane, F. D. M.

doi:10.1103/physrevlett.101.010504

Cited by 1,474 publications

(1,958 citation statements)

References 14 publications

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“…[24] (see formula (13) and (14) in that reference where we have defined the boundary perturbation as…”

Section: Discussionmentioning

confidence: 99%

“…The ES depends on the bipartition and is, by definition, the set of pseudoenergies ξ i 's [13]. Throughout our paper we will use the notation with double rightangle for kets in the 2 + 1 physical Hilbert space, while simple rightangles are reserved for states in the CFT.…”

Section: Entanglement Of the Trial Statesmentioning

confidence: 99%

“…Li and Haldane (LH) studied the ES of quantum Hall systems numerically in Ref. [13], and observed that it contains a low-lying part in which the multiplicities are related in a universal way to the ones of the conformal field the-ory (CFT) describing the low-energy edge excitations. This low-lying universal part is usually well-separated from the rest of the ES by an entanglement gap (see also the extended discussion in [14]).…”

Section: A Motivationmentioning

confidence: 99%

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Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states

2012

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We consider the trial wavefunctions for the fractional quantum Hall effect that are given by conformal blocks, and construct their associated edge excited states in full generality. The inner products between these edge states are computed in the thermodynamic limit, assuming generalized screening (i.e. short-range correlations only) inside the quantum Hall droplet, and using the language of boundary conformal field theory (boundary CFT). These inner products take universal values in this limit: they are equal to the corresponding inner products in the bulk 2d chiral CFT which underlies the trial wavefunction. This is a bulk/edge correspondence; it shows the equality between equal-time correlators along the edge and the correlators of the bulk CFT up to a Wick rotation.This approach is then used to analyze the entanglement spectrum of the ground state obtained with a bipartition A ∪ B in real-space. Starting from our universal result for inner products in the thermodynamic limit, we tackle corrections to scaling using standard field-theoretic and renormalization group arguments. We prove that generalized screening implies that the entanglement Hamiltonian HE = − log ρA is isospectral to an operator that is local along the cut between A and B. We also show that a similar analysis can be carried out for particle partition. We discuss the close analogy between the formalism of trial wavefunctions given by conformal blocks and Tensor Product States, for which results analogous to ours have appeared recently. Finally, the edge theory and entanglement spectrum of px ± ipy paired superfluids are treated in a similar fashion in the appendix.

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“…[24] (see formula (13) and (14) in that reference where we have defined the boundary perturbation as…”

Section: Discussionmentioning

confidence: 99%

Section: Entanglement Of the Trial Statesmentioning

confidence: 99%

Section: A Motivationmentioning

confidence: 99%

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Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states

2012

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“…This geometry is commonly used for accessing larger systems as the unique ground state (selected by the finite-size shift, see Appendix A) on the sphere facilitates the computation task. We analyze the topological entanglement entropy (TEE) 49,50 and the entanglement spectrum (ES) 51 in different tunneling regimes with different ground-state degeneracies on the torus, and demonstrate that they accurately match the predictions for the (331) Halperin state and the MR Pfaffian state in the weak-and intermediatetunneling regime, respectively. Importantly, all characterizations of phases are robust and stable for various system sizes [from N e = 14 to 24 (see Appendix Sec.…”

Section: Entanglement Spectroscopymentioning

confidence: 99%

Fractional quantum Hall bilayers at half filling: Tunneling-driven non-Abelian phase

et al. 2016

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Multicomponent quantum Hall systems with internal degrees of freedom provide a fertile ground for the emergence of exotic quantum liquids. Here we investigate the possibility of non-Abelian topological order in the half-filled fractional quantum Hall (FQH) bilayer system driven by the tunneling effect between two layers. By means of the state-of-the-art density-matrix renormalization group, we unveil "finger print" evidence of the non-Abelian Moore-Read Pfaffian state emerging in the intermediate-tunneling regime, including the ground-state degeneracy on the torus geometry and the topological entanglement spectroscopy (entanglement spectrum and topological entanglement entropy) on the spherical geometry, respectively. Remarkably, the phase transition from the previously identified Abelian (331) Halperin state to the non-Abelian Moore-Read Pfaffian state is determined to be continuous, which is signaled by the continuous evolution of the universal part of the entanglement spectrum, and discontinuities in the excitation gap and the derivative of the ground-state energy. Our results not only provide a "proof-of-principle" demonstration of realizing a non-Abelian state through coupling different degrees of freedom, but also open up a possibility in FQH bilayer systems for detecting different chiral p−wave pairing states.

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“…There is a special basis for the ground state manifold, the minimally entangled basis, in which each basis state |a can be identified with an anyon type a [99,102,103].…”

Section: Entanglement Invariants For the Identification Of Fqh Phasesmentioning

confidence: 99%

Competing Abelian and non-Abelian topological orders inν=1/3+1/3quantum Hall bilayers

et al. 2015

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Bilayer quantum Hall systems, realized either in two separated wells or in the lowest two sub-bands of a wide quantum well, provide an experimentally realizable way to tune between competing quantum orders at the same filling fraction. Using newly developed density matrix renormalization group techniques combined with exact diagonalization, we return to the problem of quantum Hall bilayers at filling ν = 1/3 + 1/3. We first consider the Coulomb interaction at bilayer separation d, bilayer tunneling energy ∆SAS, and individual layer width w, where we find a phase diagram which includes three competing Abelian phases: a bilayer-Laughlin phase (two nearly decoupled ν = 1/3 layers); a bilayer-spin singlet phase; and a bilayer-symmetric phase. We also study the order of the transitions between these phases. A variety of non-Abelian phases have also been proposed for these systems. While absent in the simplest phase diagram, by slightly modifying the interlayer repulsion we find a robust non-Abelian phase which we identify as the "interlayer-Pfaffian" phase. In addition to non-Abelian statistics similar to the Moore-Read state, it exhibits a novel form of bilayer-spin charge separation. Our results suggest that ν = 1/3 + 1/3 systems merit further experimental study. CONTENTS

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Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States

Cited by 1,474 publications

References 14 publications

Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states

Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states

Fractional quantum Hall bilayers at half filling: Tunneling-driven non-Abelian phase

Competing Abelian and non-Abelian topological orders inν=1/3+1/3quantum Hall bilayers

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