Abstract
We analyze some features of the entanglement entropy for an integer quantum Hall state ($\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 ${\rm III}_1$ 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.
Highlights
The idea that there is some deep connection between entropy and gravity is well-known and wellaccepted [1,2,3,4]
The Reeh-Schlieder and ConnesStormer theorems [5], coupled with the observation that the algebra of local observables should be a type III1 von Neumann algebra, tell us that entanglement is an integral part of relativistic quantum field theory [6,7,8]
If we consider only the fully filled Landau level (LLL), which is what is relevant in modeling noncommutative geometry, the only freedom in the reduced density matrices is due to a change of the separating surface in M
Summary
The idea that there is some deep connection between entropy and gravity is well-known and wellaccepted [1,2,3,4]. If we consider only the fully filled LLL, which is what is relevant in modeling noncommutative geometry, the only freedom in the reduced density matrices is due to a change of the separating surface in M. For all such cases, we will see that the spectrum of the modular operator is Rþ, as the number of states tends to infinity. In relativistic quantum field theory, the algebra of local observables is expected to be a type III1 von Neumann algebra This means that the intersection of the spectra of the modular operators over all states (or density matrices) should be Rþ [6,7]. The paper concludes with a short discussion and two Appendixes with explicit details of some of the relevant calculations
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