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On the Boundary Conformal Field Theory Approach to Symmetry-Resolved Entanglement
by Giuseppe Di Giulio, René Meyer, Christian Northe, Henri Scheppach, Suting Zhao
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Submission summary
| Authors (as registered SciPost users): | Giuseppe Di Giulio · Rene Meyer |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2212.09767v2 (pdf) |
| Date accepted: | 2023-06-14 |
| Date submitted: | 2023-05-09 09:34 |
| Submitted by: | Di Giulio, Giuseppe |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study the symmetry resolution of the entanglement entropy of an interval in two-dimensional conformal field theories (CFTs), by relating the bipartition to the geometry of an annulus with conformal boundary conditions. In the presence of extended symmetries such as Kac-Moody type current algebrae, symmetry resolution is possible only if the boundary conditions on the annulus preserve part of the symmetry group, i.e. if the factorization map associated with the spatial bipartition is compatible with the symmetry in question. The partition function of the boundary CFT (BCFT) is then decomposed in terms of the characters of the irreducible representations of the symmetry group preserved by the boundary conditions. We demonstrate that this decomposition already provides the symmetry resolution of the entanglement spectrum of the corresponding bipartition. Considering the various terms of the partition function associated with the same representation, or charge sector, the symmetry-resolved R\'enyi entropies can be derived to all orders in the UV cutoff expansion without the need to compute the charged moments. We apply this idea to the theory of a free massless boson with $U(1)$, $\mathbb{R}$ and $\mathbb{Z}_2$ symmetry.
Author comments upon resubmission
we express our gratitude for the valuable feedback that enhanced the quality of the manuscript. We have incorporated all the suggestions and criticisms raised in the two reports, including the addition of two new equations and relevant comments, primarily in section 5. In addition, we have accommodated the editor's request for more emphasis on the new findings by relegating some of the existing discussions on known results into the appendices. We have consequently modified the references to the equations, adding several sentences which stress the connection with the new appendices. We believe that this restructuring not only addresses the editor's request but also improves the coherence and clarity of the main text. Finally, we have fixed some minor typos. The details of the modifications are provided below.
List of changes
List of changes:
- We have adapted the second paragraph of section 5.3, sketching the application of the technique exploited in the present work to the non-abelian group$\hat{su}(2)_k$.
- We have added a paragraph in section 5.1 just below the bullet list, to clarify the role of the boundary states in our analysis.
- We have added a paragraph at the end of section 5.1 to point out further applications of the formalism employed in the present manuscript.
- In the final part of the third bullet point in section 5.1, we have added a paragraph for stressing the dependence of our results on the regularization.
- We have extended the discussion on the first bullet point of section 5.1 in order to compare the approach presented in the present work with other existing methods.
- We have added a paragraph ( the fourth, in the current version of the manuscript) in section 5.3 to comment on the generalization of the BCFT approach to multi-interval geometries.
- We have corrected minor typos in the following equations: (51), (53), (55), (63), (77), (103d).
- The entire section 3.2 (in the numbering of the previous version of the manuscript) is moved into appendix B of the present manuscript. Appendix B in the previous version becomes appendix C in the current version;
- The text and the equations from Eq.\,(30a) to the entire paragraph after Eq.\,(32b) of the previous version of the manuscript goes into appendix A.1 of the present manuscript (appendix A of the previous version becomes appendix A.2 in the current version).
Published as SciPost Phys. Core 6, 049 (2023)