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Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies
by YuAn Chen, PoShen Hsin
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Submission summary
Authors (as registered SciPost users):  YuAn Chen · PoShen Hsin 
Submission information  

Preprint Link:  https://arxiv.org/abs/2110.14644v2 (pdf) 
Date submitted:  20220616 04:50 
Submitted by:  Hsin, PoShen 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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Approach:  Theoretical 
Abstract
We construct infinitely many new exactly solvable local commuting projector lattice Hamiltonian models for general bosonic beyond group cohomology invertible topological phases of order two and four in any spacetime dimensions, whose boundaries are characterized by gravitational anomalies. Examples include the beyond group cohomology invertible phase without symmetry in (4+1)D that has an anomalous boundary $\mathbb{Z}_2$ topological order with fermionic particle and fermionic loop excitations that have mutual $\pi$ statistics. We argue that this construction gives a new nontrivial quantum cellular automaton (QCA) in (4+1)D of order two. We also present an explicit construction of gapped symmetric boundary state for the bosonic beyond group cohomology invertible phase with unitary $\mathbb{Z}_2$ symmetry in (4+1)D. We discuss new quantum phase transitions protected by different invertible phases across the transitions.
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Reports on this Submission
Anonymous Report 2 on 20221222 (Contributed Report)
 Cite as: Anonymous, Report on arXiv:2110.14644v2, delivered 20221221, doi: 10.21468/SciPost.Report.6364
Strengths
1. They construct explicit lattice Hamiltonian models of (4+1)D beyond cohomology invertible phases, in the case with Z2 0form symmetry or without any global symmetry. They also obtain the boundary theory given by Z2 gauge theory in (3+1)D.
2. In the case without global symmetry, the boundary (3+1)D Z2 gauge theory of the beyondcohomology invertible phase (with effective action w2w3) supports a magnetic loop excitation phrased as a 'fermonic loop' in literature. They discovered a nontrivial statistical property of the fermonic loop excitation based on the concrete lattice model and effective field theory.
Report
This paper gives concrete lattice Hamiltonian models for the beyondcohomology invertible phases in (4+1)D. In particular, their construction of the w2w3 phase without symmetry is quite clever, where they express the w2w3 phase in terms of a higherform Z2 gauge theory whose action does not have explicit dependence of StiefelWhitney classes w2, w3. This makes possible to realize a simple lattice model without employing explicit cochain representatives of w2, w3 on lattice. They also study the statistical property of the fermonic loop excitation of the (3+1)D Z2 gauge theory on its boundary, and found that its orientationreversing defect supports a fermion.
It clarifies the existing arguments about fermionic loops in literature, and regarded as an important contribution about understanding of fermionic loops. Overall, this is an excellent paper and I can safely recommend this article for publication.
Requested changes
They provide a disentangler of the lattice model for w2w3 phase, and claim that the disentangler realizes a nontrivial quantum cellular automaton (QCA) in (4+1)D. This claim should be a conjecture, based on the other primary conjecture that the (3+1)D Z2 gauge theory w/ fermonic particle and a fermonic loop cannot be realized by a commuting projector Hamiltonian.
However, I do not know if anyone studied or even talked about this conjecture, while it is widely believed that the chiral topological phase in (2+1)D cannot be realized by a commuting projector model, which leads to a construction of nontrivial QCA in (3+1)D.
I think it would benefit community, if the author can add some comment about the conjecture about commuting projector model described above. But this is an additional comment which is not necessarily required for publication. I can already recommend this article for publication.
Anonymous Report 1 on 2022819 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2110.14644v2, delivered 20220819, doi: 10.21468/SciPost.Report.5557
Strengths
1 Explicit construction of Hamiltonian model gapped boundary of beyond group cohomology SPTs.
2 The idea of integrating the Lagrange multiplier and regarding an invertible system as a finite gauge theory, then realize it as a Hamiltonian system.
Report
The manuscript describes the construction of projector hamiltonian models of gapped boundaries of SPTs that are beyond group cohomology. The idea is to realize the SPT as a finite gauge theory with a suitable DW twist, and to realize it as a Hamiltonian system. Then the gapped boundary is obtained by truncating the lattice system.
The construction as a Hamiltonian system is new and will be helpful in the study of SPT and topological phases, and thus I recommend publishing it in SciPost after the authors consider the suggestions below.
Requested changes
1 Although the construction of Hamiltonian system is new, as for a spacetime system it is discussed in https://arxiv.org/1905.05391 . The bosonic cases it is briefly discussed in Section 3. (Despite the name of the title, timereversal symmetry is not necessarily in the construction.) It would be ideal to discuss the relationship to this construction.
2 Typo in the title of section 2.1: Beyong > Beyond.
3 The sentence “If we denote the background..” on page 10 does not finish.
Author: PoShen Hsin on 20220824 [id 2749]
(in reply to Report 1 on 20220819)We thank the referee for carefully reading the manuscript and pointing out the reference.
In our construction for the invertible phase, we introduce instead a bulk dynamical gauge field b that is constrained to satisfy b = (StiefelWhitney classes) . If there is a boundary, there are different boundary conditions for b. The manuscript focused on the rough or Dirichlet boundary condition, and also briefly discussed the free boundary condition. The gapped boundary in 1905.05391 corresponds to the Dirichlet boundary condition of b, which is also discussed in the manuscript.
Anonymous on 20220825 [id 2750]
(in reply to PoShen Hsin on 20220824 [id 2749])Thank you for your comments. 1. I agree with the author's explanation, but I still think it is worth explaining in the article. 2. and 3. Thanks.
Anonymous on 20220825 [id 2753]
(in reply to Anonymous Comment on 20220825 [id 2750])Thank you for the suggestion. We will add the explanation in the article.