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  • Open Access

Mixed Phases in Feedback Ising Models

Yi-Ping Ma1,*, Ivan Sudakow2,†, and P. L. Krapivsky3,4,‡

  • *Contact author: yiping.ma@northumbria.ac.uk
  • †Contact author: ivan.sudakow@open.ac.uk
  • ‡Contact author: pkrapivsky@gmail.com
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Phys. Rev. Lett. 137, 027101 – Published 7 July, 2026

DOI: https://doi.org/10.1103/3ck1-gzzf

Abstract

We study mean-field Ising models in which the coupling depends on the magnetization via a feedback function. We identify mixed phases (MPs) and show that they can be stable at zero temperature for sufficiently strong feedback. Moreover, stable MPs are always superstable, meaning that perturbations decay linearly in time. Feedback Ising models (FIMs) provide a useful framework for phase transformations between aligned phases via stable and unstable intermediate phases in multistable systems. We also analyze the dynamical behavior of FIMs driven by a varying magnetic field and discuss basic properties of finite-dimensional FIMs.

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Phys. Rev. Lett. 137, 027101– Published 7 July, 2026
  • Received 12 June 2025
  • Accepted 19 May 2026
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