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Intrinsically/Purely GaplessSPT from NonInvertible Duality Transformations
by Linhao Li, Masaki Oshikawa, Yunqin Zheng
Submission summary
Authors (as registered SciPost users):  Li Linhao · Masaki Oshikawa 
Submission information  

Preprint Link:  https://arxiv.org/abs/2307.04788v1 (pdf) 
Date submitted:  20231016 07:24 
Submitted by:  Linhao, Li 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The KennedyTasaki (KT) transformation was used to construct the gapped symmetry protected topological (SPT) phase from the symmetry breaking phase with open boundary condition, and was generalized in our proceeding work [L. Li et al. arXiv:2301.07899] on a ring by sacrificing the unitarity, and should be understood as a noninvertible duality transformation. In this work, we further apply the KT transformation to systematically construct gapless symmetry protected topological phases. This construction reproduces the known examples of (intrinsically) gapless SPT where the nontrivial topological features come from the gapped sectors by means of decorated defect constructions. We also construct new (intrinsically) purely gapless SPTs where there are no gapped sectors, hence are beyond the decorated defect construction. This construction elucidates the field theory description of the various gapless SPTs, and can also be applied to analytically study the stability of various gapless SPT models on the lattice under certain symmetric perturbations.
Current status:
Reports on this Submission
Strengths
This is a very interesting and timely research topic.
Weaknesses
This paper describes a lot of detailed examples with little clarification of the underlying conceptual ideas.
Report
The authors address a very interesting and timely subject of topological aspects of gapless phases, which have gotten a fair bit of attention in the recent past.
The paper makes copious use of the KennedyTasaki transformation, which essentially implements a combination of gauging the Z2 x Z2 symmetry with pasting an SPT. Using this transformation and some knowledge of the c=1/2 Ising and c=1 free boson transitions, the authors are able to study various topological aspects in a doubled spin1/2 chain with Z2 x Z2 symmetry.
Despite the interesting analysis, it is the referee’s feeling that the underlying structure could be described a bit more in detail. As it is written, the authors go quickly into examples and sometimes the reader is left missing the underlying conceptual view.
A suggestion— perhaps the authors would add a section clarifying the general structure of various types of gapless SPTs. This is partially done in the introduction and particularly table 1 and eq 1.21.9, but more clean definitions of these would be welcome.
Requested changes
Some comments/questions:
1) Perhaps this is a naive question. Why is the Triv to SPT transition in 1.5 not a gapless SPT. It is obtained by stacking an Ising^2 CFT by a Z2 x Z2 SPT. Is it because the entire symmetry acts on the gapless sector. If so, it would be very nice if the authors could provide a clean definition of gapless SPTs.
2) In many places in the paper it is written Z_SSB =delta(A). Is a factor of 2 (related to the ground state degeneracy) missing?
3) Around 5.19, what is the nature of this anomaly. It is known that there are no anomalies for Z2 symmetry in bosonic systems. Perhaps these can be cancelled by some local counter terms. could the authors please comment on this ?
4) Is there a symmetry based explanation of the size dependent GSD in the intrinsically gapless SPT?
5) The section heading of Sec. 2.2 has a typo.
Report
Although the classification of gapped symmetryprotected topological (SPT) phases is by now wellestablished, the study of SPT effects in gapless systems is still an open avenue of research. Several kinds of gapless SPT orders have been introduced over the recent years, including gapless SPT, intrinsically gapless SPT, and purely gapless SPT.
The authors provide a unifying perspective on these effects, mostly focusing on spin1/2 chains with Z_2 x Z_2 symmetry. This unified description relies on the KennedyTasaki transformation, which provides a powerful approach to study these questions. The wide range of results (Eqs 1.1 to 1.9) they manage to obtain using this technique is very impressive and provides a clear physical picture for the rich physics associating gaplessness and SPT order in such systems. They also provide the first example (to the best of my knowledge) of an intrinsically purely gapless SPT.
The paper is wellwritten and very pedagogical. I thus recommend publication, provided the authors address my comment below:
• As far as I can tell, an explanation of the notation of gauge degrees of freedom is not given. For example, above Eq 3.4, Z[A_sigma,A_tau] is used but A_sigma,A_tau are not defined. For the sake of the paper being selfcontained, could the authors explain more clearly their notation for the gauging procedure, and for formulas like Eq 3.5? They also use an integration over “X_2” which is not explicitly defined I believe? I think they probably use notation they introduced in an earlier paper, but it would be good to redefine it here.