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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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Submission summary

Authors (as registered SciPost users): Philipp Schmoll
Submission information
Preprint Link: scipost_202110_00010v1  (pdf)
Date accepted: 2021-10-20
Date submitted: 2021-10-11 09:43
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.

Published as SciPost Phys. 11, 098 (2021)

Author comments upon resubmission

We thank the editor in charge for the helpful processing of our manuscript. We would also like to express our gratitude to the referees, for the evaluation and careful reading of our submission, as well as for insightful comments to improve its quality. We have addressed the feedback in a revised version.

List of changes

- the sentence about Bethe-integrability appeared in the context of the 1D classical Heisenberg model and was unfortunately fully misplaced. It was moved to the corresponding passage in the conclusion, now relating to the O(3)-NLSM and its lattice counterpart, the 1D quantum Heisenberg
- reference [Liu et al., Phys. Rev. D 88, 056005 (2013)] has been added to the bibliography

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