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Multiparticle states investigation with the tensor renormalization group method

Fathiyya Izzatun Az-zahra1,*, Shinji Takeda1,†, and Takeshi Yamazaki2,3,‡

  • *Contact author: fathiyya@hep.s.kanazawa-u.ac.jp
  • †Contact author: takeda@hep.s.kanazawa-u.ac.jp
  • ‡Contact author: yamazaki@het.ph.tsukuba.ac.jp
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Phys. Rev. D 114, 074501 – Published 2 October, 2026

DOI: https://doi.org/10.1103/zhq7-swns

Abstract

We investigate multiparticle states of the (1+1)d Ising Model using a spectroscopy scheme based on the transfer matrix and tensor renormalization group method. The scheme begins with computing the energy spectrum of the system from the transfer matrix estimated by the coarse-grained tensor network. The quantum number and momentum of these energy eigenstates are not a priori known, thus we identify them using matrix elements of an interpolating operator that is numerically computed with an impurity tensor network. Furthermore, by observing the dependence of the energy as a function of system size, we identify the number of particles of the eigenstates and obtain one-, two-, and three-particle states for a specific quantum number and momentum. From the two-particle state sector, we compute the scattering phase shift using Lüscher’s formula and the wave function approach, and observe their consistency with theoretical prediction. Using the information of the two-particle scattering phase shift, we investigate the degeneracy of the two-particle states, the theoretical prediction of the three-particle finite volume energy, and also the degeneracy in the three-particle states.

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Phys. Rev. D 114, 074501– Published 2 October, 2026
  • Received 23 June 2026
  • Accepted 21 August 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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