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The classical two-dimensional Heisenberg model revisited: An $SU(2)$-symmetric tensor network study

by Philipp Schmoll, Augustine Kshetrimayum, Jens Eisert, Roman Orus, Matteo Rizzi

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Authors (as registered SciPost users): Philipp Schmoll
Submission information
Preprint Link:  (pdf)
Date submitted: 2021-06-29 08:59
Submitted by: Schmoll, Philipp
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Computational
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational


The classical Heisenberg model in two spatial dimensions constitutes one of the most paradigmatic spin models, taking an important role in statistical and condensed matter physics to understand magnetism. Still, despite its paradigmatic character and the widely accepted ban of a (continuous) spontaneous symmetry breaking, controversies remain whether the model exhibits a phase transition at finite temperature. Importantly, the model can be interpreted as a lattice discretization of the $O(3)$ non-linear sigma model in $1+1$ dimensions, one of the simplest quantum field theories encompassing crucial features of celebrated higher-dimensional ones (like quantum chromodynamics in $3+1$ dimensions), namely the phenomenon of asymptotic freedom. This should also exclude finite-temperature transitions, but lattice effects might play a significant role in correcting the mainstream picture. In this work, we make use of state-of-the-art tensor network approaches, representing the classical partition function in the thermodynamic limit over a large range of temperatures, to comprehensively explore the correlation structure for Gibbs states. By implementing an $SU(2)$ symmetry in our two-dimensional tensor network contraction scheme, we are able to handle very large effective bond dimensions of the environment up to $\chi_E^\text{eff} \sim 1500$, a feature that is crucial in detecting phase transitions. With decreasing temperatures, we find a rapidly diverging correlation length, whose behaviour is apparently compatible with the two main contradictory hypotheses known in the literature, namely a finite-$T$ transition and asymptotic freedom, though with a slight preference for the second.

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Reports on this Submission

Anonymous Report 2 on 2021-9-9 (Contributed Report)

  • Cite as: Anonymous, Report on arXiv:2106.06310v2, delivered 2021-09-09, doi: 10.21468/SciPost.Report.3512


In this paper, the authors apply the tensor network algorithm to study possible phase transitions in 2D Heisenberg model. Although the result was not clear enough to resolve the main controversy, it pushed the boundary of numerical simulation on this problem and revealed interesting features. It could provide a useful reference for future bigger scale simulations. The paper is nicely written, with careful explanation of many details of the algorithm. I recommend publication as it is.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Author:  Philipp Schmoll  on 2021-10-11  [id 1833]

(in reply to Report 2 on 2021-09-09)

Thank you very much for your careful reading, informative summary and positive comments. We appreciate your assessment of our results and the recommendation for publication.

Anonymous Report 1 on 2021-8-12 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2106.06310v2, delivered 2021-08-12, doi: 10.21468/SciPost.Report.3382


In this manuscript, the authors reported a large-scale tensor network numerical study of the two-dimensional (2D) classical Heisenberg model with O(3) symmetry.

The classical Heisenberg model is a paradigmatic model in statistical physics and condensed matter. While extensive analytical and numerical efforts were devoted to this model in the past, it remains a controversial issue whether the 2D classical O(3) Heisenberg model on the square lattice exhibits a Berezinskii-Kosterlitz-Thouless (BKT) phase transition at finite temperature.

Taking advantage of their tensor network library incorporating SU(2) symmetry, the authors address this important open problem. After casting the partition function into a tensor network, a symmetry-preserving corner transfer matrix renormalization group (CTMRG) approach is employed to (approximately) contract the tensor network in the thermodynamic limit. Within this framework, various physical quantities, including spin-spin correlation functions, thermodynamic entropy, and entanglement measures, have been computed. In particular, a rapidly diverging correlation length is observed at low temperature, which is consistent with the two contradictory hypotheses in the literature: finite-temperature BKT transition and asymptotic freedom (i.e., absence of finite-temperature transition).

With decreasing temperatures, the truncation of angular momentum basis in the partition function causes increasingly large errors. However, this is unavoidable for carrying out CTMRG contractions. It is then extremely difficult to conclusively answer whether the transition temperature is exactly zero or very low. Thus, this leaves room for future investigations.

In my opinion, the results reported in this manuscript surely represent state-of-the-art tensor network calculations and certainly shed new light on an important open problem. The manuscript is very well written. It would also motivate further development of tensor network methods and applying them to other paradigmatic models. Therefore, I would warmly recommend its publication in SciPost.

Below I have two minor comments:

1) In page 8, the claim that “the model is Bethe integrable” is a bit confusing. As far as I know, the computation of the partition function for the one-dimensional classical Heisenberg model relies on (directly) diagonalizing the transfer matrix in the angular momentum basis, so the Bethe ansatz technique is not used here, nor in Refs. [71,72].

2) In page 9, the tensor network representation of the classical O(3) Heisenberg model reminds me of an earlier paper [Liu et al., Phys. Rev. D 88, 056005 (2013)], where a similar tensor network representation was derived in Sec. III B. If they coincide, it might make sense to add a reference.

  • validity: top
  • significance: high
  • originality: high
  • clarity: high
  • formatting: perfect
  • grammar: excellent

Author:  Philipp Schmoll  on 2021-10-11  [id 1834]

(in reply to Report 1 on 2021-08-12)

Thank you very much for your careful reading, informative summary and positive comments. We appreciate your assessment of our results and the recommendation for publication.

Below we provide responses to the comments point-by-point.

1) We apologize for the confusion, the sentence is unfortunately fully misplaced. It should have appeared in the context of the 1D quantum Heisenberg model (and the associated 1D O(3) NLSM), which are Bethe integrable. 2) Thank you for pointing us to the reference. It is indeed of relevance to our study since it is an equivalent re-formulation of the Boltzmann factors.

We have addressed your feedback in a revised version of the manuscript, with the following changes.

1) We have moved the sentence regarding Bethe-integrability to the corresponding passage in the conclusion. 2) The reference [Liu et al., Phys. Rev. D 88, 056005 (2013)] has been added to the Bibliography.

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