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Tensor renormalization group approach to critical phenomena via symmetry-twisted partition functions

Shinichiro Akiyama1,2,*, Raghav G. Jha3,†, Jun Maeda4,‡, Yuya Tanizaki5,§, and Judah Unmuth-Yockey6,7,∥

  • *Contact author: akiyama@ccs.tsukuba.ac.jp
  • †Contact author: raghav.govind.jha@gmail.com
  • ‡Contact author: maeda@gauge.scphys.kyoto-u.ac.jp
  • §Contact author: yuya.tanizaki@yukawa.kyoto-u.ac.jp
  • ∥Contact author: jfunmuthyockey@gmail.com
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Phys. Rev. D 113, 074502 – Published 2 April, 2026

DOI: https://doi.org/10.1103/ywhr-lt6w

Abstract

The locality of field theories strongly constrains the possible behaviors of symmetry-twisted partition functions, and thus they serve as order parameters to detect low-energy realizations of global symmetries, such as spontaneous symmetry breaking (SSB). We demonstrate that the tensor renormalization group (TRG) offers an efficient framework to compute the symmetry-twisted partition functions, which enables us to detect the symmetry-breaking transition and also to study associated critical phenomena. As concrete examples of SSB, we investigate the two-dimensional (2D) classical Ising model and the three-dimensional (3D) classical O(2) nonlinear sigma model, and we identify their critical points solely from the twisted partition function. By employing the finite-size scaling argument, we find the critical temperature Tc=2.2017(2) with the critical exponent ν=0.663(33) for the 3D O(2) model. In addition, we also study the Berezinskii–Kosterlitz–Thouless (BKT) criticality of the 2D classical O(2) model by extracting the helicity modulus from the twisted partition functions, and we obtain the BKT transition temperature, TBKT=0.8928(2).

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Phys. Rev. D 113, 074502– Published 2 April, 2026
  • Received 7 January 2026
  • Accepted 11 March 2026
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