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Effective Fractonic Behavior in a TwoDimensional Exactly Solvable Spin Liquid
by Guilherme Delfino, Weslei B. Fontana, Pedro R. S. Gomes, Claudio Chamon
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Submission summary
Authors (as registered SciPost users):  Guilherme Delfino · Weslei Fontana 
Submission information  

Preprint Link:  https://arxiv.org/abs/2207.00409v2 (pdf) 
Date submitted:  20220711 14:38 
Submitted by:  Delfino, Guilherme 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
In this work we propose a $\mathbb{Z}_N$ clock model which is exactly solvable on the lattice. We find exotic properties for the lowenergy physics, such as UV/IR mixing and excitations with restricted mobility, that resemble fractonic physics from higher dimensional models. We then study the continuum descriptions for the lattice system in two distinct regimes and find two qualitative distinct field theories for each one of them. A characteristic time scale that grows exponentially fast with $N^2$ (and diverges rapidly as a function of system parameters) separates these two regimes. For times below this scale, the system is described by an effective fractonic ChernSimonslike action, where higherform symmetries prevent quasiparticles from hoping. In this regime, the system behaves effectively as a fracton as isolated particles, in practice, never leave their original position. Beyond the large characteristic time scale, the excitations are mobile and the effective field theory is given by a pure mutual ChernSimons action. In this regime, the UV/IR properties of the system are captured by a peculiar realization of the translation group.
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Reports on this Submission
Anonymous Report 2 on 2022819 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2207.00409v2, delivered 20220819, doi: 10.21468/SciPost.Report.5562
Strengths
1solid, 2comprehensive
Report
The authors gave a solid and comprehensive study on a model whose excitations have restricted mobility. In the exact solvable limit, there are subextensive ground state degeneracy which is a function of system size. There are excitations only allowed to move along certain directions. Then, the authors formulated an effective field theory to describe the fractonic phase and also the phase where all the excitations turn into mobile. Although I think the paper meets the standard of SciPost, I am still confused about a few points. In order to better understand the results, I list my questions in the following and hope the author can make further clarification.
1. Can the lattice model in Eq.2 be obtained by Higgsing a U(1) model/theory?
2. In the effective theory, there is an immobile defect (Eq.47). It looks like a Wilson loop along time direction. Is it an instanton? Can I understand it in the language of the lattice model?
3. As the defect carries the charge of a global higher form symmetry, is this symmetry an exact microscopic (UV) symmetry of the lattice model? If so, what is the symmetry transformation in terms of lattice operators? How can I see it leaves the lattice Hamiltonian invariant?
4. By tuning the perturbation in Eq.29 very large, all the excitations gets full mobile. Can we understand this phase transition by a picture of the anyon condensation?
Anonymous Report 1 on 2022818 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2207.00409v2, delivered 20220818, doi: 10.21468/SciPost.Report.5555
Report
The authors present a systematic analysis of a 2+1D lattice model that arises as a projection of the 3+1D Chamon code onto the plane. Consequently, the model displays features that are qualitatively similar to those of fracton models. The authors also identify two distinct regimes that are separated by an exponentially longtime scale and are governed by different effective actions. The results are presented clearly and the paper is generally wellwritten.
This is an interesting paper that should be published once the following (mostly minor) points are addressed:
1. While the model itself is new, its physics is that of a symmetryenriched topological (SET) phase where lattice translations permute inequivalent anyon types. Although the authors do not make this identification explicitly, viewed thusly the positiondependent braiding statistics and latticesize dependent ground state degeneracy are not surprising. I would suggest that the authors make reference to the existing literature and situate their model in this broader context.
2. The authors do not state what topological order the model possesses; it is clearly $\mathbb{Z}_N^2$, but this should be mentioned somewhere.
3. Similarly, how many distinct anyon types are present in the model? There should be $N^4$, which would be consistent with the field theory, but the authors should directly verify this in the lattice model.
4. While the action of translation symmetry on the anyons is discussed, how do the two mirror symmetries $M_x$ and $M_y$ act? Are there any other lattice symmetries that act nontrivially on the anyons?
5. With regards to the effective fractonic ChernSimons theory, the authors should cite the following two papers where a similar action was first written down in terms of symmetric tensor gauge fields:
Phys. Rev. B 96, 125151 (2017) and Phys. Rev. B 97, 085116 (2018)
6. The authors should cite SciPost Phys. 10, 011 (2021) and mention Ref.[22] in the conclusions when discussing the existence of intrinsically fracton phases in 2+1D (under mild assumptions, both offer proofs that this is not possible).
7. Finally, the idea of realizing "effective" fractonic behavior for exponentially longtime scales has previously been discussed theoretically and even demonstrated experimentally both in one and two dimensions (although the mechanism differs from that proposed by the authors). See e.g., Phys. Rev. X 10, 011042 (2020).
Author: Guilherme Delfino on 20221006 [id 2887]
(in reply to Report 1 on 20220818)
We thank the referee for the questions, suggestions, and references. We addressed these points below.
1. While the model itself is new, its physics is that of a symmetryenriched topological (SET) phase where lattice translations permute inequivalent anyon types. Although the authors do not make this identification explicitly, viewed thusly the positiondependent braiding statistics and latticesize dependent ground state degeneracy are not surprising. I would suggest that the authors make reference to the existing literature and situate their model in this broader context.
Thank you for the suggestion. We added a brief discussion about this point in the introduction of the paper.
2. The authors do not state what topological order the model possesses; it is clearly $ \mathbb{Z}_N^2$ but this should be mentioned somewhere.
We now state what topological order the model possesses explicitly in the end of section II.B.
3. Similarly, how many distinct anyon types are present in the model? There should be $N^4$, which would be consistent with the field theory, but the authors should directly verify this in the lattice model.
In the lattice model, there are as many distinct anyons as degenerate ground states $ \dim \mathcal{H}_0 $. For the case in which $ L_x=L_y $ are multiple of $N$, this reduces to the usual $ N^4 $ result. For the general case, however, a more complicated relation for the number of anyons arises according to Eq. (12), a number bounded by $ N $ and $ N^4 $. Although the topological order is $ \mathbb{Z}_N\times \mathbb{Z}_N $, the number of independent quasiparticles is reduced because of the existence of the $ N $hopping line operators.
4. While the action of translation symmetry on the anyons is discussed, how do the two mirror symmetries $M_x$ and $M_y$ act? Are there any other lattice symmetries that act nontrivially on the anyons?
In the paper we have only considered the translation operations that act as identity on the anyon space, as they are relevant in the analysis of topological properties, as GSD and statistical angles. We believe that an investigation of the lattice symmetries action on anyons would require an approach similar to that in arXiv:2204.07111, where the authors introduce anyons that are explicitly dependent on position. Addressing the second question, the answer is positive: there are discrete lattice symmetries that act nontrivially on the anyons, as rotations and mirror symmetries e.g. $ C_4:\mathfrak{p}_x\rightarrow \mathfrak{p}_y $ and $ M_x:\mathfrak{p}_x\rightarrow \overline{\mathfrak{p}}_x$. In the paper, however, we did not consider such analysis as it is independent of the topological properties of the system, which is the main focus of the paper.
5. With regards to the effective fractonic ChernSimons theory, the authors should cite the following two papers where a similar action was first written down in terms of symmetric tensor gauge fields: Phys. Rev. B 96, 125151 (2017) and Phys. Rev. B 97, 085116 (2018)
Thank you for the references. We have included them in the revised version of the manuscript.
6. The authors should cite SciPost Phys. 10, 011 (2021) and mention Ref.[22] in the conclusions when discussing the existence of intrinsically fracton phases in 2+1D (under mild assumptions, both offer proofs that this is not possible).
Thank you for the reference. We have included the references in our discussion in the conclusions.
7. Finally, the idea of realizing "effective" fractonic behavior for exponentially longtime scales has previously been discussed theoretically and even demonstrated experimentally both in one and two dimensions (although the mechanism differs from that proposed by the authors). See e.g., Phys. Rev. X 10, 011042 (2020).
Thank you for calling our attention to this point. We included the suggested reference, as well as a reference to ArXiv:2009.05577.
Author: Guilherme Delfino on 20221006 [id 2888]
(in reply to Report 2 on 20220819)We thank the referee for the questions. We addressed the points below.
1. Can the lattice model in Eq.2 be obtained by Higgsing a U(1) model/theory?
2. In the effective theory, there is an immobile defect (Eq.47). It looks like a Wilson loop along time direction. Is it an instanton? Can I understand it in the language of the lattice model?
3. As the defect carries the charge of a global higher form symmetry, is this symmetry an exact microscopic (UV) symmetry of the lattice model? If so, what is the symmetry transformation in terms of lattice operators? How can I see it leaves the lattice Hamiltonian invariant?
4. By tuning the perturbation in Eq. (29) very large, all the excitations gets full mobile. Can we understand this phase transition by a picture of the anyon condensation?