SciPost Submission Page
Pivoting through the chiral-clock family
by Nick G. Jones, Abhishodh Prakash, Paul Fendley
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Nick Jones |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2406.01680v1 (pdf) |
| Date submitted: | 2024-06-12 13:26 |
| Submitted by: | Jones, Nick |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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Abstract
The Onsager algebra, invented to solve the two-dimensional Ising model, can be used to construct conserved charges for a family of integrable $N$-state chiral clock models. We show how it naturally gives rise to a "pivot" procedure for this family of chiral Hamiltonians. These Hamiltonians have an anti-unitary CPT symmetry that when combined with the usual $\mathbb{Z}_N$ clock symmetry gives a non-abelian dihedral symmetry group $D_{2N}$. We show that this symmetry gives rise to symmetry-protected topological (SPT) order in this family for all even $N$, and representation-SPT (RSPT) physics for all odd $N$. The simplest such example is a next-nearest-neighbour chain generalising the spin-1/2 cluster model, an SPT phase of matter. We derive a matrix-product state representation of its fixed-point ground state along with the ensuing entanglement spectrum and symmetry fractionalisation. We analyse a rich phase diagram combining this model with the Onsager-integrable chiral Potts chain, and find trivial, symmetry-breaking and (R)SPT orders, as well as extended gapless regions. For odd $N$, the phase transitions are "unnecessarily" critical from the SPT point of view.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1- interesting set of models
2- good calculations
3- timely topic
Weaknesses
1- the presentation is not appropriate, as the paper is too hard to read and not precise enough about the crucial concept it discusses.
Report
In this paper, the authors consider a family of integrable Hamiltonians that are formed out of linear combinations of (a representation of) Onsager algebra generators. They show that these are related to each other by a “pivot” procedure, and that this therefore helps finding Hamiltonians in the “SPT” phase.
The paper is interesting and seems to produce good results and nice and fundamental ideas. The idea of using the Onsager algebra in order to study the Hamiltonians and especially their SPT property is very interesting. I believe this paper can be published in Scipost Physics in some form.
However in the current form I find the paper very difficult to read. It lacks the basic definitions of some of the main concepts discussed (like “SPT phase” itself); the general discussion in the introduction and section 2 are only marginally useful, and the reader must try to guess what is happening as the more precise results for the specific models are then discussed. I don’t have specific comments on the physics or mathematics, which I believe is correct and deep. But I believe the presentation should be very greatly improved.
The authors should provide much better explanations about what are “pivot relation” and “SPT phase”, and how a Hamiltonian can be “in a phase” (instead of a state being in a phase, as is usual in statistical mechanics). See the comments below for more (related and non-related) questions and requirement for improved explanations.
Introduction: What does “any Hamiltonians giving rise to this algebra” mean? What does it mean that a Hamiltonian “gives rise” to an algebra? Perhaps that the Hamiltonian lies within this algebra (a specific representation of it)? Also, it seems that the statement that any two Hamiltonians satisfy “the same pivot relations” is not so clear. I thought the pivot relation was to produce SPT phases, but not all cases do produce SPT phases (from the abstract, and from the rest of the paper). What is, then, really, a “pivot relation”? This is also not clear at this stage.
Page 3: the statement that (2) results in an infinite-dimensional algebra as stated (with two discrete families) is in fact not very meaningful. Its only meaning is that the Dolan-Grady relations are not enough to make the algebra finite-dimensional. Indeed any two generators, without additional relations, lead to an algebra that is, at most, countably-dimensional, which can be organised into any number of discrete families as one wishes. Could the author make the statement more precise? Perhaps define the $A_l$ and $G_m$, or give their main properties? Or just refer to (4) already?
Page 4, beginning of 2.2: what is the meaning of “trivial ground state”? Please give some explanations there.
Page 4, after eq (6): could the author explain in some words with basic equations what SPT order means for $H_{SPT}$? What is the “SPT order”? What is “an SPT Hamiltonian”? How is the ground state of $H_{SPT}$ non-trivial? What is an “SPT” entangler”? What is the meaning of a Hamiltonian “being a non-trivial SPT phase”? This is strange for somebody coming from standard statistical mechanics, as normally the terminology “phase” is reserved for certain thermodynamic states of a given Hamiltonian, and not for characterising a Hamiltonian. More explanations about these fundamental concepts would make the paper more readable.
Page 4, 2.3: what is the meaning of “behave nicely” here? Please be more precise.
Page 5, after eq 12: why are the two families related by KW duality? Have the authors prove this? More explanations would be good.
Page 5, paragraph around eq 13: it is not clear to me how phase structures are obtained from these relations. The explanation is not so convincing: that all $A_{2k}$ are non-trivial SPT phases seem a strong requirement, and the conclusion that $A_k/4$ is a pivot seems then like a tautology.
Page 9, beginning of sect 4: “we showed in section 3.4 that…” What “such a form” does the ground stat of $A_2$ have? The unitary transformation of a “trivial” hamiltonian (sum of commuting terms, etc)? Please be more specific - I don’t think SPT phase was shown already.
Page 9, bottom of the page: it is not clear (at that point) what is meant by a representation of symmetry generators “in the ground state”. The ground state is a single state (or do the authors mean degeneracies? The ground space?), and is usually not enough to form by itself a representation. Do the authors mean the representation on the MPS form of the ground state, within the equivalence class of MPS matrices for that state?
Requested changes
See the report
Recommendation
Ask for major revision
Strengths
1-Detailed study of the phase diagrams for the Hamiltonian $H(\alpha,\beta,\gamma)$ based on a variety of numerical probes
Weaknesses
2-The notions of SPT and RSPT could be introduced more clearly
Report
This is a very interesting and complete study of the physics of a family of $N$ states chiral clock models based on the Onsager algebra. A first observation is that the latter can be used as a pivot between the different Onsager generators, enabling one to construct the ground state of generators $A_l$ from the known ones in the cases $l=0,1$. Since the pivot procedure acts as an SPT entangler, some of the constructed Hamiltonians have SPT order (or a weaker form thereof known as RSPT).
The authors then move on to a detailed numerical investigation of the phase diagram of Hamiltonians constructed from $A_0,A_1,A_2$, generalising a known result for $N=2$. The phase diagrams display a rich variety of transitions between gap, gapless and (R)SPT phases.
Despite some blind spots in the phase diagram (some of which are the subject of a forthcoming companion paper), I think this work decisively meets the criteria for publication in SciPost. I however have some comments, listed below. In particular, it feels that the notions of SPT and RSPT could be better introduced for someone not familiar with the subject. Since the case $N=2$ is well-known and an excellent toy-model for the SPT physics, I would suggest to start Section 4 with a brief review of the $N=2$ physics. In particular, it would be useful to see the ground state of $A_2$ written explicitly, in order to present a first concrete encounter with symmetry fractionalisation.
Requested changes
1- Section 2: all the pivots considered here map between $A_m$ generators. Is there a reason why the authors did not consider pivots between the generators $G_m$ ?
2- As explained above, I suggest to start Section 4 with a brief review of the known $N=2$ physics, introducing in particular the spin-1/2 cluster model, SPT and symmetry fractionalisation
3- Looking at the phase diagram for $N=2,3,4$ (Fig. 3), one is lead to wonder whether the gapless region keeps on expanding for larger $N$. Could the authors comment ? Is there any insight about the $N\to \infty$ limit?
Minor requests :
4- above eq. (23), in the definition of $\mathcal{V}$ : is $\mathcal{K}$ inside or outside the product ? Please add parentheses to resolve the ambiguity.
5- there is a typo after eq. (29) : "The $N$ ground states of $E_1$" : $E_1$ should be replaced by $A_1$.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)