arXiv:2006.10920 · Physical review. D/Physical review. D.

Entanglement entropy for integer quantum Hall effect in two and higher dimensions

Dimitra Karabali A

Published Jul 21, 2020• 8 citations

Abstract

We analyze the entanglement entropy, in real space, for the higher-dimensional integer quantum Hall effect on ${\mathbb{C}\mathbb{P}}^{k}$ (any even dimension) for Abelian and non-Abelian magnetic background fields. In the case of $\ensuremath{\nu}=1$ we perform a semiclassical calculation which gives the entropy as proportional to the phase-space area. This exhibits a certain universality in the sense that the proportionality constant is the same for any dimension and for any background, Abelian or non-Abelian. We also point out some distinct features in the profiles of the eigenfunctions of the two-point correlator that underline the difference in the value of entropies between $\ensuremath{\nu}=1$ and higher Landau levels.

DOI: 10.1103/physrevd.102.025016

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arXiv:2006.10920 · Physical review. D/Physical review. D.

Entanglement entropy for integer quantum Hall effect in two and higher dimensions

Dimitra Karabali A

Published Jul 21, 2020• 8 citations

arXiv HTML PDF

Abstract

We analyze the entanglement entropy, in real space, for the higher-dimensional integer quantum Hall effect on

{\mathbb{C}\mathbb{P}}^{k}

(any even dimension) for Abelian and non-Abelian magnetic background fields. In the case of

\ensuremath{\nu}=1

we perform a semiclassical calculation which gives the entropy as proportional to the phase-space area. This exhibits a certain universality in the sense that the proportionality constant is the same for any dimension and for any background, Abelian or non-Abelian. We also point out some distinct features in the profiles of the eigenfunctions of the two-point correlator that underline the difference in the value of entropies between

\ensuremath{\nu}=1

and higher Landau levels.

DOI: 10.1103/physrevd.102.025016

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We analyze the entanglement entropy, in real space, for the higher-dimensional integer quantum Hall effect on

{\mathbb{C}\mathbb{P}}^{k}

(any even dimension) for Abelian and non-Abelian magnetic background fields. In the case of

\ensuremath{\nu}=1

\ensuremath{\nu}=1

and higher Landau levels.

\mathcal{N}

N = 4 Yang Mills theory on S3. In this latter context, the entanglement we consider is an entanglement in target space. We comment on possible implications to gauge-gravity duality.

\left({GSD}_{\tilde{A}}\right)

GSD A ˜ , where

{GSD}_{\tilde{A}}

GSD A ˜ is the ground state degeneracy of a particular restriction of the full model to A. The quantity

{GSD}_{\tilde{A}}

GSD A ˜ can be further divided into a contribution that scales with the size of the boundary ∂A and a term which depends on the topology of ∂A. There is also a topological contribution coming from A itself, that may be non-zero when A has a non-trivial homology. We present some examples and discuss how the topology of A affects the topological entropy. Our formalism allows us to do most of the calculation for arbitrary dimension d. The result is in agreement with entanglement calculations for known topological models.

d

, of the quasi-two-dimensional system for a number of filling fractions $\ensuremath{\nu}$ in the lowest and the second Landau levels: $\ensuremath{\nu}$ = 1/3, 7/3, 1/2, and 5/2. We observe that the entanglement measures are dependent on which Landau level the electrons fractionally occupy, and find that filling fractions 1/3 and 7/3, which are considered to be Laughlin states, weaken with

d

in the lowest Landau level ($\ensuremath{\nu}=1/3$) and strengthen with

d

in the second Landau level ($\ensuremath{\nu}=7/3$). For the enigmatic even-denominator $\ensuremath{\nu}=5/2$ state, we find that entanglement in the ground state is consistent with that of the non-Abelian Moore-Read Pfaffian state at an optimal thickness,

d

. We also find that the single-layer $\ensuremath{\nu}=1/2$ system is not a fractional quantum Hall state consistent with the experimental observation. In general, our theoretical findings based on entanglement considerations are completely consistent with the results based on wave-function overlap calculations.