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Coexistence of valence-bond formation and topological order in the Frustrated Ferromagnetic $J_1$-$J_2$ Chain

by Cliò Efthimia Agrapidis, Stefan-Ludwig Drechsler, Jeroen van den Brink, Satoshi Nishimoto

This Submission thread is now published as SciPost Phys. 6, 019 (2019)

Submission summary

As Contributors: Cliò Efthimia Agrapidis · Satoshi Nishimoto · Jeroen van den Brink
Arxiv Link: (pdf)
Date accepted: 2019-01-23
Date submitted: 2018-12-10 01:00
Submitted by: Agrapidis, Cliò Efthimia
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


Frustrated one-dimensional (1D) magnets are known as ideal playgrounds for new exotic quantum phenomena to emerge. We consider an elementary frustrated 1D system: the spin-$\frac{1}{2}$ ferromagnetic ($J_1$) Heisenberg chain with next-nearest-neighbor antiferromagnetic ($J_2$) interactions. On the basis of density-matrix renormalization group calculations we show the existence of a finite spin gap at $J_2/|J_1|>1/4$ and we find the ground state in this region to be a valence bond solid (VBS) with spin-singlet dimerization between third-neighbor sites. The VBS is the consequence of spontaneous symmetry breaking through order by disorder. Quite interestingly, this VBS state has a Affleck-Kennedy-Lieb-Tasaki-type topological order. This is the first example of a frustrated spin chain in which quantum fluctuations induce gapped topological order.

Ontology / Topics

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Ferromagnetic phases Heisenberg spin chains

Published as SciPost Phys. 6, 019 (2019)

Author comments upon resubmission

Dear Editor,
We resubmit our paper titled "Coexistence of valence-bond formation and topological order in the Frustrated Ferromagnetic $J_1$-$J_2$ Chain". We have addressed the referee comments, adding new results on entanglement spectra and performing minor revisions. A list of changes is visible below.

Kind regards,
C. E. Agrapidis, S.-L. Drechsler, J. van den Brink, S. Nishimoto

List of changes

-Appendices B-C moved to Sec. 3 Spin gap.
-Appendices D-F moved to Sec. 5 Dimerization order.
-Fig. 1 changed to make the represented orders more clear and to include previous Fig. 9(b).
-New Fig. 2 includes the figures from App. B.
-New Fig. 3 is similar to previous Fig. 2: the inset is now Fig. 3(c) and we include the finite size scaling for the spin gap in Fig. 3(b).
-New Fig. 5 shows results for the entanglement spectra of the system.
-New Fig. 6 includes previous Fig. 9(a) and 4(a).
-New Fig. 7 includes previous Fig. 4(b) and 10.
-New Fig. 9 includes previous Fig. 11 and 12.
-More details about the use of the denomination "order by disorder" have been added in the introduction.
-An explanation of the statement "arbitrary set of valence bonds" has been added to the manuscript.
-The cause for the two-fold degeneracy of the ground state, namely the ferromagnetic dimerization, is now explicitly addressed.
-To support our claim of topological order in the $\mathcal D_3$-VBS state, we have included new results about entanglement spectra in Sec. 4 Valence Bond Solid.
-We added a footnote to explain the formula $4-2(\delta \bmod 2)$ for the symmetry breaking period.
-We changed the references to Supplemental Material to be to App. A or App. B
-We added an explanation on how the phase transition works in Sec. 7 Conclusion to better put our findings into context.
-We added a short discussion about the consequences of our theoretical findings for real materials in Sec. 7 Conclusion.

Reports on this Submission

Anonymous Report 2 on 2018-12-14 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1806.08662v4, delivered 2018-12-14, doi: 10.21468/SciPost.Report.745


1- solid numerical investigation of the ferromagnetic Heisenberg chain with S=1/2 supplemented by frustrating antiferromagnetic coupling between next-nearest neighbors.
2-leading edge DMRG code
3-nice results on the tiny spin gaps in this system.
4- kind of VBS ordering is sorted out.


- slight overselling of the results


The manuscript has improved so that I am inclined to recommend its publication.
However, one of the two main selling points, order by disorder,
are still not well corroborated by the results.
What is the classical degeneracy of the system? I do not see any except for
fine-tuned parameters J1, J2 and J3, but not for generic sets of these parameters.
(The occurrence of a gap in a dimerized AFM chain as such destroys the LRO,
but it does not imply any "order by disorder" mechanism.)

Requested changes

I do not request changes, but I recommend to give the authors
the opportunity to substantiate their one of their main claims, see Report.

  • validity: good
  • significance: good
  • originality: good
  • clarity: ok
  • formatting: good
  • grammar: good

Anonymous Report 1 on 2018-12-12 (Invited Report)


See earlier report


See earlier report.


The authors have implemented my suggestions. I feel that the manuscript is in better shape now and can be publsihed.

  • validity: good
  • significance: -
  • originality: ok
  • clarity: good
  • formatting: excellent
  • grammar: excellent

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