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Fractal Symmetric Phases of Matter
by Trithep Devakul, Yizhi You, F. J. Burnell, S. L. Sondhi
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|As Contributors:||Trithep Devakul|
|Arxiv Link:||https://arxiv.org/abs/1805.04097v2 (pdf)|
|Date submitted:||2018-05-25 02:00|
|Submitted by:||Devakul, Trithep|
|Submitted to:||SciPost Physics|
We study spin systems which exhibit symmetries that act on a fractal subset of sites, with fractal structures generated by linear cellular automata. In addition to the trivial symmetric paramagnet and spontaneously symmetry broken phases, we construct additional fractal symmetry protected topological (FSPT) phases via a decorated defect approach. Such phases have edges along which fractal symmetries are realized projectively, leading to a symmetry protected degeneracy along the edge. Isolated excitations above the ground state are symmetry protected fractons, which cannot be moved without breaking the symmetry. In 3D, our construction leads additionally to FSPT phases protected by higher form fractal symmetries and fracton topologically ordered phases enriched by the additional fractal symmetries.
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Anonymous Report 1 on 2018-9-7 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1805.04097v2, delivered 2018-09-07, doi: 10.21468/SciPost.Report.571
This paper has imported many important ideas in symmetry-protected topological order with global symmetry to systems with subsystem/fractal symmetries.
I can believe most of the arguments, but those on protected edge modes are not at a similar level of rigor as the other parts of the paper. See Requested changes.
This paper studies quantum phase of matter that has a symmetry group generated by operators on a fractal subsystem. Important ideas from SPT literature such as robust edge modes, symmetry twist, and duality (gauging) are imported to this setting, and combined with certain fundamental immobility of excitations due to fractal symmetries. I believe this combination is a nontrivial contribution to literature.
The language of polynomials they use may seem unusual, but is somewhat inevitable since the underlying fractals are generated by linear cellular automata where polynomials are prevalent. I appreciate the authors' effort to remain in conventional language as much as possible.
* Among many sections of good exposition, there is one section that comes short. In Sec. 5.3, the basis of the argument is the nontriviality of the projective representation induced at the edge by the two symmetry operators that commute in the bulk. This basis alone is solid; in 1D SPT with a global internal symmetry, one can consider manifestly commuting two subgroups of the full symmetry group, inspect the induced symmetry action on the edge, and find that those at one edge is nontrivially projective. Here, it is crucial that two subgroups manifestly commuted in the bulk. I don't think this is the case for the symmetry described in page 15. The anticommuting pair of factors of the symmetry operator might be compensated by some near-edge tensor factors, and it is not clear, at least not explained in the manuscript, whether such near-edge compensation is irrelevant. Without this step, the discussion of projective representation is ill-founded.
* Some minor suggestions:
* The word symmetry is used for two technically different things. In the manuscript, sometimes it means the symmetry group, and sometimes it is an element of the symmetry group, and sometimes it means generator of some subgroup of the symmetry group. I managed to figure out which means which from the context, but it would read better if carefully and technically the three were distinguished. There are some standard way of saying it. Instead of "number of symmetries", one typically speak of the order of the symmetry group, for example.
* Around Eq.(23) the correlation function decays at a rate of volume of the fractal object (with Hausdorff dimension) but lacks any further explanation. Why is it the case?
* Below Eq.(26) it is not controlled-Z but is contolled-X in the present basis.
* "equal superposition" => equal amplitude superposition.
* On page 15, the second to the last paragraph, "faithful representation" is a technical and well-established term in the representation theory, but the usage here is improper. The proper wording is "linear representation", or one could simply say "usual representation." The faithfulness means the injectivity from the group under consideration into a matrix group where the former is represented into.