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Free fermion entanglement with a semitransparent interface: the effect of graybody factors on entanglement islands
by Jorrit Kruthoff, Raghu Mahajan, Chitraang Murdia
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|Authors (as registered SciPost users):
|Jorrit Kruthoff · Raghu Mahajan
We study the entanglement entropy of free fermions in 2d in the presence of a partially transmitting interface that splits Minkowski space into two half-spaces. We focus on the case of a single interval that straddles the defect, and compute its entanglement entropy in three limits: Perturbing away from the fully transmitting and fully reflecting cases, and perturbing in the amount of asymmetry of the interval about the defect. Using these results within the setup of the Poincaré patch of AdS$_2$ statically coupled to a zero temperature flat space bath, we calculate the effect of a partially transmitting AdS$_2$ boundary on the location of the entanglement island region. The partially transmitting boundary is a toy model for black hole graybody factors. Our results indicate that the entanglement island region behaves in a monotonic fashion as a function of the transmission/reflection coefficient at the interface.
Published as SciPost Phys. 11, 063 (2021)
Author comments upon resubmission
List of changes
Comments for Referee #1:
1) The axial and vector phases are allowed for in the general S matrix (2.6). The phases do not show up in the EE computations. This is because Eq (A.47) is general, and it only contains |c_1|^2 and |c_2|^2.
2) The quantities r, t and \gamma are dimensionless, and so our expressions for the entropy and the QES location are valid when the expansion parameter is the smallest relevant dimensionless quantity. In particular Eq (3.9) is valid when t is smallest dimensionless parameter in the problem, and so the case t >> k/b is outside the regime of validity of (3.9).
3) While we do not have a proof, we suspect that the behavior of the QES as a function of the boundary conditions goes the opposite way when the black hole is evaporating, where we are using the setup of Ref  for an evaporating black hole. This depends a bit on the details of how the evaporating black hole is setup. We have added comments about this in the last paragraph.
Comments for Referee #2:
1) The divergence as L_+ or L_- go to zero is a manifestation of the UV divergent nature of EE. If L_- is strictly zero, one should go to the beginning of the replica setup and start the computation again. One can see this in a simple example in flat space with a fully reflecting boundary, and for the interval [L_-, L_+] with both L_- and L_+ positive. The answer is 1/6 Log (4 L_+ L_- (L_+ - L_-)^2 / (L_+ + L_-)). The limit L_->0 is not smooth. The issue is that the highly UV modes are relevant here.
2) We have commented on the flat-flat vs AdS-flat interfaces after equation (3.3). There is a contribution to the EE from the Weyl factor in AdS. We have clarified this further after Eq (3.3).
3) The T>0 case is beyond the scope of our paper. The difficulty is that in the T>0 case studied in Ref.  there will now be two interfaces, one for left bath and one for the right bath. We have not tried to solve this more difficult problem.
4) We have added a couple of sentences to the last paragraph of the paper highlighting some interesting avenues for future work. There is also a sentence about future work at the end of section 2. We have consciously decided not to have a separate section for future directions, since the future directions for section 2 and section 3 are quite different and more appropriately discussed in each section individually.
Submission & Refereeing History
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Reports on this Submission
The authors' response and modifications to the manuscript have addressed the concerns in report 1, in particular emphasizing the impact of the choice of vacuum in the evaporating black hole and clarifying the limits in section 3. I believe the manuscript is now ready for publication.