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Entanglement in the quantum Hall fluid of dipoles
by Jackson R. Fliss
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Submission summary
As Contributors:  Jackson Fliss 
Preprint link:  scipost_202107_00003v2 
Date accepted:  20210826 
Date submitted:  20210806 17:49 
Submitted by:  Fliss, Jackson 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We revisit a model for gapped fractonic order in (2+1) dimensions (a symmetrictraceless tensor gauge theory with conservation of dipole and tracequadrupole moments described in \cite{Prem:2017kxc}) and compute its groundstate entanglement entropy on $\mathbb R^2$. Along the way, we quantize the theory on open subsets of $\mathbb R^2$ which gives rise to gapless edge excitations that are Lifshitztype scalar theories. We additionally explore varieties of gaugeinvariant extended operators and rephrase the fractonic physics in terms of the local deformability of these operators. We explore similarities of this model to the effective field theories describing quantum Hall fluids: in particular, quantization of dipole moments through a novel compact symmetry leads us to interpret the vacuum of this theory as a dipole condensate atop of which dipoles with fractionalized moments appear as quasiparticle excitations with Abelian anyonic statistics. This interpretation is reflected in the subleading ``topological entanglement" correction to the entanglement entropy. We extend this result to a series of models with conserved multipole moments.
Published as SciPost Phys. 11, 052 (2021)
Author comments upon resubmission
List of changes
1) added a sentence (below equation (40)) defining $\mathcal N (\delta[]'s)$
2) added a footnote (footnote 7) explaining that the Jacobian from changing pathintegral variables A_{ij}>\phi is canceled by an inverse determinant from the delta function constraint.
3) Added explicit parameterizations to figure 2(a)(f) and modified caption of figure 2 so that it is clear that the examples satisfy the conditions for gauge invariance.
4) Expanded the first paragraph of section 2.1 to clarify to the reader for when results hold for R^2 and when they hold on general surface. Added sentence to footnote 6 to explain how quantization on T^2 differs and added relevant reference.
5) Updated first paragraph of section 4.1 to mention the prior work on crystal melting as suggested by referee 2.
6) Updated second paragraph of page 4 to mention prior work noting quantum Hall physics and tensor gauge theory as suggested by referee 2.
7)Moved Appendix A (previous draft) to main text as section 5.2 as suggested by referee 1. I have given the bulk gauge theory calculation its own heading ("5.1 The extended Hilbert space") to better distinguish the two approaches.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 1 on 202188 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202107_00003v2, delivered 20210808, doi: 10.21468/SciPost.Report.3352
Report
The author replied that there is no typo in equation (43), then I don't understand the meaning of $q^a(\partial \bar{\partial})^2(\mathbf{z}_a ,\mathbf{z}_b)q^b$.
Regarding examples in Figs (2e) and (2f), I have my last question to the author. Could the cases happen in a lattice? Do they violate the quantization of dipole moment? Of course, my question doesn't affect the claims of the manuscript since the author begins discussing the quantization in section 4.
Other than that I am satisfied with the author's answers. I recommend the paper to be published on Scipost Physics.
Author: Jackson Fliss on 20210808 [id 1645]
(in reply to Report 1 on 20210808)Thank you to the anonymous report for the insightful questions. With regards to these questions:
1) On equation (43): $(\partial\bar{\partial})^{2}$ is a distribution (the inverse of $(\partial\bar{\partial})^2$) which takes two spatial points as its argument. These are evaluated at the locations of the defects (${\bf z}_a$) because the source, $\rho$, appearing in the first line of equation (43) is localized to the point of the defects.
2) On figures (2e) and (2f): I do not think these strip operators can, strictly speaking, be realized on a lattice although some approximation to them is possible depending on the details of the lattice. If so then I do think their lattice approximation will not violate dipole quantization.