{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/tensor-network-renormalization-study-on-the","title":"Tensor Network Renormalization Study on the Crossover in Classical Heisenberg and $\\mathrm{RP^2}$ Models in Two Dimensions","arxiv_id":"2202.07042","date":"2022-02-14","proceeding":null,"authors":["Atsushi Ueda","Masaki Oshikawa"],"abstract":"We study the classical two-dimensional $\\mathrm{RP^2}$ and Heisenberg models, using the Tensor-Network Renormalization (TNR) method. The determination of the phase diagram of these models has been challenging and controversial, owing to the very large correlation lengths at low temperatures. The finite-size spectrum of the transfer matrix obtained by TNR is useful in identifying the conformal field theory describing a possible critical point. Our results indicate that the ultraviolet fixed point for the Heisenberg model and the ferromagnetic $\\mathrm{RP^2}$ model in the zero temperature limit corresponds to a conformal field theory with central charge $c=2$, in agreement with two independent would-be Nambu-Goldstone modes. On the other hand, the ultraviolet fixed point in the zero temperature limit for the antiferromagnetic Lebwohl-Lasher model, which is a variant of the $\\mathrm{RP^2}$ model, seems to have a larger central charge. This is consistent with $c=4$ expected from the effective SO(5) symmetry. At $T >0$, the convergence of the spectrum is not good in both the Heisenberg and ferromagnetic $\\mathrm{RP^2}$ models. Moreover, there seems no appropriate candidate of conformal field theory matching the spectrum, which shows the effective central charge $c \\sim 1.9$. These suggest that both models have a single disordered phase at finite temperatures, although the ferromagnetic $\\mathrm{RP^2}$ model exhibits a strong crossover at the temperature where the dissociation of $\\mathbb{Z}_2$ vortices has been reported.","url_abs":"https://arxiv.org/abs/2202.07042v3","url_pdf":"https://arxiv.org/pdf/2202.07042v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"tensor-network-renormalization-study-on-the","repo_url":"https://github.com/dartsushi/loop-tnr_rgflow","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}