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

Abstract

We analyze some features of the entanglement entropy for an integer quantum Hall state (ν ¼ 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 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.

Entanglement Entropy and Phase Space Density: Lowest Landau Levels and 1/2 BPS states

“…There have been many works on entanglement entropy of integer quantum Hall states using the connection to noncommutative geometry and Chern Simons theory, see e.g. [47] and references therein 8 . It would be good to understand the connection of these approaches to ours, which directly deals with the fermionic field theory.…”

Entanglement Entropy and Matter-Gravity Couplings for Fuzzy Geometry

“…I will not go over the derivation of this formula since it has appeared in the literature before [10] and has been used in related work [1,11] and in the previous talk [12] to calculate the entanglement entropy for quantum Hall droplets.…”

“…In this talk, I will present two results related to the use of fuzzy geometry as the underlying structure for a theory of gravity. This is based on the work published in [1] and [2]. Let me begin by recalling that in fuzzy geometry we have an -dimensional Hilbert space of states H , which maybe viewed as describing the (dynamical) degrees of freedom pertaining to space itself.…”

“…The dominant term for the EE will be, as usual, proportional to the area of the interface. One can even see that there are some features reminiscent of the type III 1 von Neumann algebra for the local observables [1]. But here my focus will be on the dependence of this EE on the gauge fields and spin connection in the emergent continuum.…”

“…But here my focus will be on the dependence of this EE on the gauge fields and spin connection in the emergent continuum. We will argue that this dependence is given by a generalized Chern-Simons (CS) form [1]. The interest in the question of how EE may be related to the gauge fields and spin connection and how it can inform the issue of gravity for fuzzy spaces are due to the following observations.…”