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arXiv:2001.04957 · Physical review. D/Physical review. D.

Entanglement for quantum Hall states and a generalized Chern-Simons form

V. P. Nair

Published Jun 26, 2020• 4 citations
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Abstract

We analyze some features of the entanglement entropy for an integer quantum Hall state ($\ensuremath{\nu}=1$) in comparison with ideas from relativistic field theory and noncommutative geometry. The spectrum of the modular operator, for a restricted class of states, is shown to be similar to the case of field theory or a type ${\mathrm{III}}_{1}$ von Neumann algebra. We present arguments that the main part of the dependence of the entanglement entropy on background fields and geometric data such as the spin connection is given by a generalized Chern-Simons form. Implications of this result for bringing together ideas of noncommutative geometry, entropy and gravity are briefly commented upon.

DOI
10.1103/physrevd.101.125021

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We analyze some features of the entanglement entropy for an integer quantum Hall state (\ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1) in comparison with ideas from relativistic field theory and noncommutative geometry. The spectrum of the modular operator, for a restricted class of states, is shown to be similar to the case of field theory or a type III1{\mathrm{III}}_{1}III1​ von Neumann algebra. We present arguments that the main part of the dependence of the entanglement entropy on background fields and geometric data such as the spin connection is given by a generalized Chern-Simons form. Implications of this result for bringing together ideas of noncommutative geometry, entropy and gravity are briefly commented upon.
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In the framework of Algebraic Quantum Field Theory, several operator algebraic notions of entanglement entropy can be associated with any pair of causally disjoint spacetime regions SA\mathcal{S}_ASA​ and SB\mathcal{S}_BSB​ with positive relative distance. Among them, the canonical entanglement entropy is defined as the von Neumann entropy of a canonical intermediate type I factor. In this work, we show that the canonical entanglement entropy of the vacuum state is finite for a broad class of conformal nets including the U(1)U(1)U(1)-current model and the SU(n)SU(n)SU(n)-loop group models. Since previous studies suggest that this finiteness property is related to nuclearity properties of the system, we show that the mutual information is finite in any local QFT satisfying a modular ppp-nuclearity condition for some 0<p<10 < p < 10<p<1. A similar finiteness result is established for another notion of entanglement entropy introduced in this paper. We conclude with remarks for future work in this direction.