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Entanglement in the quantum Hall fluid of dipoles
by Jackson R. Fliss
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Submission summary
| Authors (as registered SciPost users): | Jackson Fliss |
| Submission information | |
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| Preprint Link: | scipost_202107_00003v1 (pdf) |
| Date submitted: | 2021-07-02 14:13 |
| Submitted by: | Fliss, Jackson |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We revisit a model for gapped fractonic order in (2+1) dimensions (a symmetric-traceless tensor gauge theory with conservation of dipole and trace-quadrupole moments described in \cite{Prem:2017kxc}) and compute its ground-state 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 Lifshitz-type scalar theories. We additionally explore varieties of gauge-invariant 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 quasi-particle 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.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2021-8-4 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202107_00003v1, delivered 2021-08-04, doi: 10.21468/SciPost.Report.3340
Strengths
The topics are interesting. Most of the calculations are presented in detail. The author covered most of the questions of quantizing the CS theory of traceless symmetric tensor gauge on R^3.
Weaknesses
There are still some technical steps that are missed.
Report
This manuscript proposed a path integral quantization procedure for the Chern-Simon action of a version of the higher-rank gauge theory, the traceless symmetric tensor gauge. The author also studied in detail the topological defects and proposed the quantization condition for multipole charges.
The topics of this manuscript are intriguing and timely, providing that the higher-rank symmetry attracts a vast of attention from both condensed matter and high energy communities. I would recommend this manuscript published on SciPost Physics if the author could clarify the following issues/questions satisfactorily.
1) There is no explicit definition of $\mathcal{N}(\delta[]' s)$ in the manuscript.
2) When changing the path-integral variable from $A_{ij}$ to $\phi$ (From Eq (36) to Eq (43)), do we need to worry about the explicit form of the Jacobian, or the Jacobian was automatically taken care of?
3) I can see the examples strip operators in Figs 2(a)-2(d) satisfy the conditions (71) and (73). Could the author demonstrate explicitly that examples 2(e) and 2(f) also meet those conditions?
4) Extra: The manuscript proposed a quantization procedure for CS theory of a higher-rank gauge on a general space $\Sigma$, but only the results for $\Sigma \sim R^2$ are derived explicitly. Can one extend the calculations to non-trivial topological space (Torus $T^2$, for example)?
Other comments:
A) The restoration of mobility of fractonic excitations was studied previously in the context of fracton/elasticity duality in SciPost Phys. 9, 076 (2020), Phys. Rev. B 100, 045119 (2019), Phys. Rev. Lett. 121, 235301 (2018). In the dual picture, the condensation of dipoles (dislocations) corresponds to quantum melting. One can choose to condense dislocations (dipoles) in one direction to achieve the smectic phase or condense dislocations (dipoles) in both direction to obtain the nematic phase.
B) The direct connection between quantum Hall physics and the traceless symmetric tensor gauge theory was proposed in arXiv:2103.09826. One can use the higher-rank gauge symmetry to analyze the fractonic behaviours of low energy excitations in Fractional Quantum Hall systems.
Minor:
There could be a typo in equation (43)
Report #1 by Anonymous (Referee 1) on 2021-7-21 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202107_00003v1, delivered 2021-07-20, doi: 10.21468/SciPost.Report.3271
Report
The present manuscript is a technical work, where rank-2 Chern-Simons theory is quantized on a plane and then the entanglement entropy is calculated using a method that goes back to the work of Buividovich and Polikarpov. In this method the Hilbert space of a gauge theory is enlarged to include non-gauge invariant states, but has simple, tensor product structure.
The first four sections of the work, is revision and repackaging of mostly known results and serves as a nice introduction to a reader not familiar with the field. Particularly nice is discussion of the large gauge transformations.
The main new result is presented in section 5. I did not attempt to reproduce the calculation of the entanglement entropy however, I expect that it is correct. What gives me confidence is the relation of the answer to the known result for the EE of the U(1)_k Chern-Simons theory. On a plane rank-2 theory can be written as U(1)_k Chern-Simons theory with two gauge fields. This was pointed out in Ref. [1] of the manuscript. Thus the naïve answer for the EE would be twice the EE of U(1)_k Chern-Simons theory, which appears to be the case and was shown rigorously.
The sub-leading correction is the well-known topological entanglement entropy and is determined by the quantum dimension D, S = logD. In the present case D = k^{1/2} * k^{1/2} which give S = log(k). Consequently I believe it is correct as well, and also is shown rigorously.
Finally, I would like to highlight the calculation of the EE using the edge theory, which in the present case is non-relativistic Lifshitz theory. I think this calculation is important and, perhaps, should be moved to the main text. The reason is the following. While formally one might guess (in view of analogy between two copies of U(1)_k CS theory and rank-2 CS theory) that the edge theory is two chiral bosons, in reality the edge theory is one ordinary scalar boson and one non-relativistic chiral boson. Still, the two theories give the same EE, which is not obvious a priori.