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arXiv:0811.2188 · Physical Review B

Entanglement entropy of integer quantum Hall states

Ivan Dario Jimenez Rodriguez, Germán Sierra

Published Oct 7, 2009• 70 citations
arXivHTMLPDF

Abstract

We compute the entanglement entropy, in the real space, of the ground state of the integer Quantum Hall states for three different domains embedded in the cylinder, the disk and the sphere. We establish the validity of the area law with a vanishing value of the topological entanglement entropy. The entropy per unit length of the perimeter depends on the filling fraction, but it is independent of the geometry.

DOI
10.1103/physrevb.80.153303

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We compute the entanglement entropy, in the real space, of the ground state of the integer Quantum Hall states for three different domains embedded in the cylinder, the disk and the sphere. We establish the validity of the area law with a vanishing value of the topological entanglement entropy. The entropy per unit length of the perimeter depends on the filling fraction, but it is independent of the geometry.
The entanglement entropy of the incompressible states of a realistic quantum Hall system is studied by direct diagonalization. The subdominant term to the area law, the topological entanglement entropy (which is believed to carry information about topologic order in the ground state), was extracted for filling factors $\ensuremath{\nu}=1/3$, $\ensuremath{\nu}=1/5$, and $\ensuremath{\nu}=5/2$. The results for $\ensuremath{\nu}=1/3$ and $\ensuremath{\nu}=1/5$ are consistent with the topological entanglement entropy for the Laughlin wave function. The $\ensuremath{\nu}=5/2$ state exhibits a topological entanglement entropy consistent with the Moore-Read wave function.
Generally speaking, entanglement entropy (EE) between two subregions of a gapped quantum many-body state is proportional to the area/length of their interface due to the short-range quantum correlation. However, the so-called area law is violated logarithmically in a quantum critical phase. Moreover, the subleading correction exists in long-range entangled topological phases. It is referred to as topological EE which is related to the quantum dimension of the collective excitation in the bulk. Furthermore, if a non-smooth sharp angle is in the presence of the subsystem boundary, a universal angle dependent geometric contribution is expected to appear in the subleading correction. In this work, we simultaneously explore the geometric and edge contributions in the integer quantum Hall (IQH) state and its edge reconstruction in a unified bipartite method. Their scaling is found to be consistent with conformal field theory (CFT) predictions and recent results of particle number fluctuation calculations.
The entanglement entropy of the incompressible states of a realistic quantum Hall system in the second Landau level is studied by direct diagonalization. The subdominant term of the area law, the topological entanglement entropy, which is believed to carry information about topological order in the ground state, was extracted for filling factors ν = 12/5 and ν = 7/3. While it is difficult to make strong conclusions about ν = 12/5, the ν = 7/3 state appears to be very consistent with the topological entanglement entropy for the k = 4 Read–Rezayi state. The effect of finite thickness corrections to the Coulomb potential used in the direct diagonalization is also systematically studied.
We analyze the entanglement entropy, in real space, for the higher-dimensional integer quantum Hall effect on CPk{\mathbb{C}\mathbb{P}}^{k}CPk (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.