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  • Letter
  • Open Access

Numerical extraction of crosscap coefficients in microscopic models for (2+1)D conformal field theory

Jia-Ming Dong1,*, Yueshui Zhang2,*, Kai-Wen Huang1, Hong-Hao Tu2,†, and Ying-Hai Wu1,‡

  • *These two authors contributed equally.
  • †Contact author: h.tu@lmu.de
  • ‡Contact author: yinghaiwu88@hust.edu.cn
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Phys. Rev. D 112, L121701 – Published 3 December, 2025

DOI: https://doi.org/10.1103/qcp6-v1dr

Abstract

Conformal field theory (CFT) can be placed on disparate space-time manifolds to facilitate investigations of their properties. For (2+1)-dimensional [(2+1)D] theories, one useful choice is the real projective space RP3 obtained by identifying antipodal points on the boundary sphere of a three-dimensional ball. One-point functions of scalar primary fields on this manifold generally do not vanish and encode the so-called crosscap coefficients. These coefficients also manifest on the sphere as the overlaps between certain crosscap states and CFT primary states. Taking the (2+1)D Ising CFT as a concrete example, we demonstrate that crosscap coefficients can be extracted from microscopic models. We construct crosscap states in both lattice models defined on polyhedrons and continuum models in Landau levels, where the degrees of freedom at antipodal points are entangled in Bell-type states. By computing their overlaps with the eigenstates of many-body Hamiltonians, we obtain results consistent with those from conformal bootstrap. Importantly, our approach directly reveals the absolute values of crosscap overlaps, whereas bootstrap calculations typically yield only their ratios. Furthermore, we investigate the finite-size scaling of these overlaps and their evolution under perturbations away from criticality.

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Phys. Rev. D 112, L121701– Published 3 December, 2025
  • Received 25 September 2025
  • Accepted 19 November 2025
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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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