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Thermodynamics at negative temperatures in interacting systems: One-dimensional Ising model

J. Ricardo de Sousa1 and J. A. Plascak2,3,4,5

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APS Open Sci. 1, 000148 – Published 22 September, 2026

DOI: https://doi.org/10.1103/77b9-d1jl

Abstract

An exact microcanonical analysis of the one-dimensional Ising model is presented with emphasis on the emergence of negative temperatures. The number of states of the system is obtained as a function of the energy, the magnetization, and the number of spins. The Boltzmann and Gibbs entropy is computed, and the results are compared in the thermodynamic limit and for finite systems as well. According to the Boltzmann scenario, negative temperatures naturally appear as a consequence of the energy bounded spectrum. In this case, the Boltzmann entropy is not symmetric with the energy as soon as the magnetization is different from zero and does not vanish at the maximum allowed value of the energy. Along with entropy, the specific heat and magnetic susceptibility are analyzed as functions of the temperature. Although the specific heat is always positive, the symmetry is also broken in both branches of positive and negative temperatures for finite values of the magnetization. The susceptibility, in its turn, changes sign in the negative sector. In addition, the phase transition, which is present when the temperature goes to zero from the right, is totally suppressed when the temperature tends to zero from the left. Finite-size effects are also studied to understand the equivalence of the microcanonical and canonical formulations for this interacting model. Negative temperatures are not allowed according to the Gibbs formulation.

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APS Open Sci. 1, 000148– Published 22 September, 2026
Vol. 1
  • Received 6 June 2026
  • Accepted 9 September 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.

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