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A tensor network study of the complete ground state phase diagram of the spin-1 bilinear-biquadratic Heisenberg model on the square lattice

by Ido Niesen, Philippe Corboz

Submission summary

As Contributors: Ido Niesen
Arxiv Link:
Date submitted: 2017-07-13
Submitted by: Niesen, Ido
Submitted to: SciPost Physics
Domain(s): Theor. & Comp.
Subject area: Condensed Matter Physics - Theory


Using infinite projected entangled pair states, we study the ground state phase diagram of the spin-1 bilinear-biquadratic Heisenberg model on the square lattice directly in the thermodynamic limit. We find an unexpected partially nematic partially magnetic phase in between the antiferroquadrupolar and ferromagnetic regions. Furthermore, we describe all observed phases and discuss the nature of the phase transitions involved.

Current status:

Editor-in-charge has requested revision

Invited Reports on this Submission

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Anonymous Report 2 on 2017-8-24


1) Interesting system

2) Useful to use the IPEPS method, the results appear to be reliable

3) New phases are identified

4) The paper is well organized


1) Some lack of discussion of the findings (see report)


The paper is interesting and I consider the numerical results reliable. In my opinion the most interesting aspect is that the authors identify a "Haldane" phase - a 2D phase claimed to be adiabatically connected to the ground state of the spin-1 chain (the conventional Haldane state). This is also where I think some more discussion is required. The Haldane phase in 1D is adiabatically connected to the AKLT state, which can be pictorially described as the S=1 spins on a chain being "split up" into S=1/2 objects forming on-site triplets. The S=1/2 objects on nearest-neighbor sites form singlets (before the projection to triplets on the sites - after the projection the bonds consist of mixed singlets and triplets, with no S=2 components). This kind of AKLT state also exists in 2D, but with S=2 spins. It is not clear how this kind of state should be understood in a system of S=1 spins (how would the valence bonds form? some kind of resonating bonds?). Naively, it appears that the nature of the state must be quite different and it should not be adiabatically connected to the Haldane chain. This naive thinking may very well be wrong but I think the authors have to discuss the nature of this state and, if possible, present some more results supporting their claims.

Requested changes

1) Discuss the nature of the Haldane phase as discussed above and present more results supporting the claims.

  • validity: high
  • significance: high
  • originality: good
  • clarity: high
  • formatting: good
  • grammar: excellent

Anonymous Report 1 on 2017-8-24


1) Complete the phase diagram of a relevant spin 1 model on the square lattice using relatively well established tensor network techniques.
2) It unveils a new phase
3) It analyses the nature of the observed phase transitions.
4) Complements and confirms the previous results
5) It performs a rigorous analysis of the systematic approximations involved in the numerical method


1) Does not use the best available algorithms in a region where competing states belonging to different phases have very close energies ( they differ at most by 0.1% in Fig 8.).

2) It uses a lot of jargon rendering the paper fully accessible only to experts in the sector.


The paper presents solid results obtained with iPEPS on the phase diagram of the spin 1 bilinear bi-quadratic Heisenberg model.
The model is parametrized in terms of an angle \theta so that the full phase diagram is contained in the 0 2\pi interval.
The paper tries to summarize the previous results on the topic.
It points out that the region of the phase diagram between \pi and 2 \pi is known from quantum Monte Carlo studies, while the region between 0 and \pi has been investigated either approximately Ref. [4] or on small systems with ED Ref. [5]. The special point where \theta = pi/4 had also already been studied in Ref [6]. Furthermore in Ref. [21] the authors have unveiled a region of parameters in which the model exhibits an Haldane phase.

The present work seems to settle the last piece of the puzzle by identifying a new intermediate nematic partially polarized phase close to pi/2.

The numerical analysis is sound and an accurate study as a function of the bond dimension, including an extrapolation is performed to ensure that the results are correct (see Fig. 8) (On this point though I have a main concern and I would appreciate further clarifications on the chosen algorithm).

For this reason I believe it constitutes an important and sound contribution . Once my main concern has been addressed I recommend the paper for publication.

Detailed report.

The authors have tried to write a self consistent paper, but I have the impression that it could be improved by adding some pedagogical sentences here and there. I provide here some personal suggestions in that direction.

I find that the strategy of presenting the results after an historical review make the reading more difficult, I would suggest starting with an introduction to the model and then a section with the main result and the full phase diagram of Fig. 7. This would allow explaining the nature of the various phases before. I would postpone the review about what was already known prior to this paper and how the various phases have been identified.

Specific remarks following the structure of the paper


There is quite a bit of jargon, I would expect the introduction to be accessible to non-experts, though this is not a requirement, but I would suggest to avoid jargon as much as possible at this stage and try to properly define everything.
E.g. what do the authors mean for classical ground states at the beginning of pag. 2? Do they refer to configurations of three dimensional vectors, one per site (e.g. product states).

After or before Eq. 1 I would define the matrices S_i.
I would briefly define what a nematic phase means in this context something like: a phase in which the spin rotational symmetry is broken to rotations of arbitrary angles around one axis, and of pi along any axis in the perpendicular plane (if my uderstanding of the subsequent section is correct).
In particular, I do not see the connection with the nematic phases I am familiar with in liquid crystals or superconductors where I understand them as a breaking of the spatial rotational symmetries rather than the spin rotational symmetries.

-The model

At the end of pag. 4 the quadrupolar states are defined as those for which the quadrupolar moment does not vanish, I would add a definition of the quadrupolar moment.

In the following discussion I would mention that these states will be later represented as circles in the Figures such as Fig. 3, if I have understood correctly.

I would define what a time reversal operator does, this would allow to understand why the base introduced in Eq. 2 is time-reversal invariant.

I would mention after Eq. 3 that the Gell-Mann matrices are the generators of SU(3) or at least provide the relevant references.

In Eq. 4 I would directly insert J_S and J_Q and define them below as already done.

I would explain why the form of the Hamiltonian in terms of the Gell-Mann matrices is SU(3) invariant. A sentence like, an element of SU(3) is represented by a vector in a 8 dimensional space spanned by the Gell-Mann matrices, as a consequence the norm of the vector is invariant under rotations inside this space, representing SU(3) operators... (or something along this line).

-Previous studies

I would move the section further down the paper since it is a review of old results.

In Fig. 2 I would mark the SU(3) invariant points.

End of pag. 6, I would avoid referring to a part of the circle as a "corner", a circle does not have corners, maybe one could say the upper right quadrant?

Figure 3, I would expand the label explaining what the symbols and colours mean. (Maybe one could add a plot with the types of symbols (if I understand correctly the circles are quadrupolar configurations the ovals are nematic, and arrows are magnetic... but what is the colour encoding, x,z,y?).

At the end of pag. 7 3\pi /2 is listed as an SU(3) symmetric point while at pag. 5 it is not. I would add it at pag. 5.

The choice of the determinant of Q as an invariant seems strange since it would vanish as soon as one of the eigenvalues vanishes, that I do not feel means that there is no quadrupolar order, am I wrong? IIQ seems more reasonable. But I would have chosen the average of the abs of the eigenvalues, and their dispersion, in order to provide a more direct information about the magnitude of Q.
What is the advantage of the choice made in the paper?

At pag. 9 the authors mention a hint from the simple update of the existence of the AFM3, can they elaborate more (as done in the following page) or alternatively postpone the discussion completely.

-Description of all the Phases.

Second line I think is opposite directions rather than direction.

Second last line there is an extra to.

In the discussion at pag. 10 there is a part in which it is claimed that the results of the simulations with a unit cell 1x1 give lower energies than larger unit cells, how is it possible?
Is it a sign that the results with larger unit cells are not appropriately converged?

Half through page 10 I would change further investigation to further investigations.

The main results of the paper are presented in the second last paragraph of pag. 12, I would try to bring it forward as already mentioned.

#### Only important question that I expect to be answered

One of the authors (PC) has shown in recent papers that the energy of the iPEPS wave functions can be significantly improved by using new variational minimization algorithm [arXiv:1605.03006 and arXiv:1606.09170v2]. I wonder why such new algorithms have not been applied in the present scenario. It seems important in view of the presence of competing states belonging to different phases whose energy determines the phase of the system. I wonder if the authors have evidences that the variational minimization improves uniformly all competing states, or if they can certify that even a variational optimization would not further improve the energies found in the present study. It seems like a natural test, that could have been performed at least on a couple of points.


A final question concerns the extrapolation performed in Figs. 9-14. From Ref. [49] and Fig. 8 I would have expected an extrapolation as a function of the truncation error. I guess that the data are clear enough that this would not make any difference, but I am curious to understand why the authors have chosen to extrapolate as a function of D, after funding a better strategy. Is it related to the lack of scaling of Q as a function of the truncation error?

I hope that the above observations will help improving the already nice manuscript.

Requested changes

Convincing comments about weakness 1).

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

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