Complex-temperature properties of the two-dimensional Ising model for nonzero magnetic field

Victor Matveev, Robert Shrock

Research output: Contribution to journalArticlepeer-review

57 Scopus citations


We study the complex-temperature phase diagram of the square-lattice Ising model for nonzero external magnetic field H, i.e., for 0≤μ≤∞, where μ=[Formula Presented]. We also carry out a similar analysis for -∞≤μ≤0. The results for the interval -1≤μ≤1 provide a way of continuously connecting the two known exact solutions of this model, viz., at μ=1 (Onsager, Yang) and μ=-1 (Lee and Yang). Our methods include numerical calculations of complex-temperature zeros of the partition function and an analysis of low-temperature series expansions. For real nonzero H, the inner branch of a limaçon bounding the FM phase breaks and forms two complex-conjugate arcs. We study the singularities and associated exponents of thermodynamic functions at the endpoints of these arcs. For μ<0, there are two line segments of singularities on the negative and positive u axis, and we carry out a similar study of the behavior at the inner endpoints of these arcs, which constitute the nearest singularities to the origin in this case. Finally, we also determine the exact complex-temperature phase diagrams at μ=-1 on the honeycomb and triangular lattices and discuss the relation between these and the corresponding zero-field phase diagrams.

Original languageEnglish (US)
Pages (from-to)254-267
Number of pages14
JournalPhysical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Issue number1
StatePublished - 1996
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics


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