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Pivoting through the chiral-clock family

by Nick G. Jones, Abhishodh Prakash, Paul Fendley

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Submission summary

Authors (as registered SciPost users): Nick Jones
Submission information
Preprint Link: scipost_202501_00018v1  (pdf)
Date accepted: 2025-03-06
Date submitted: 2025-01-11 14:20
Submitted by: Jones, Nick
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • Condensed Matter Physics - Theory
  • Mathematical Physics

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

List of changes

Section 1:
Footnote 1 added to clarify SPT notions. Edits to the text in order to clarify pivot procedure and RSPT.

Section 2:
Clarification of Onsager algebra definition, pivoting, and KW duality, see reply to Referee 2.

Section 4:
New section 4.1 added to introduce symmetry fractionalisation in the cluster model. Surrounding material rearranged to allow this and to improve clarity.

Minor typos fixed and references added.

Published as SciPost Phys. 18, 094 (2025)


Reports on this Submission

Report #2 by Anonymous (Referee 2) on 2025-2-26 (Invited Report)

Report

The authors have mostly answered my questions. I find there is still a lot of "jargon", but this is the nature of the subject, and the additional explanations have clarified a few aspects. I recommend publication.

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Report #1 by Anonymous (Referee 1) on 2025-2-10 (Invited Report)

Report

The authors have satisfactorily addressed all the comments in my previous report. I have no further comment and recommend this version for publication in SciPost Physics.

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