Two-State Spin Systems with Negative Interactions

Authors Yumou Fei , Leslie Ann Goldberg, Pinyan Lu

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Author Details

Yumou Fei
  • School of Mathematical Sciences, Peking University, China
Leslie Ann Goldberg
  • Department of Computer Science, University of Oxford, UK
Pinyan Lu
  • Laboratory of Interdisciplinary Research of Computation and Economics (SUFE), Ministry of Education, Shanghai University of Finance and Economics, China


We thank Mingji Xia for many very helpful conversations about this work.

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Yumou Fei, Leslie Ann Goldberg, and Pinyan Lu. Two-State Spin Systems with Negative Interactions. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 45:1-45:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We study the approximability of computing the partition functions of two-state spin systems. The problem is parameterized by a 2×2 symmetric matrix. Previous results on this problem were restricted either to the case where the matrix has non-negative entries, or to the case where the diagonal entries are equal, i.e. Ising models. In this paper, we study the generalization to arbitrary 2×2 interaction matrices with real entries. We show that in some regions of the parameter space, it’s #P-hard to even determine the sign of the partition function, while in other regions there are fully polynomial approximation schemes for the partition function. Our results reveal several new computational phase transitions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Approximate Counting
  • Spin Systems
  • #P-Hardness
  • Randomized Algorithms


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