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Exact Potts model partition functions on ladder graphs

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  • Shrock, Robert

Abstract

We present exact calculations of the partition function Z of the q-state Potts model and its generalization to real q, for arbitrary temperature on n-vertex ladder graphs, i.e., strips of the square lattice with width Ly=2 and arbitrary length Lx, with free, cyclic, and Möbius longitudinal boundary conditions. These partition functions are equivalent to Tutte/Whitney polynomials for these graphs. The free energy is calculated exactly for the infinite-length limit of these ladder graphs and the thermodynamics is discussed. By comparison with strip graphs of other widths, we analyze how the singularities at the zero-temperature critical point of the ferromagnet on infinite-length, finite-width strips depend on the width. We point out and study the following noncommutativity at certain special values qs:limn→∞limq→qsZ1/n≠limq→qslimn→∞Z1/n. It is shown that the Potts antiferromagnet on both the infinite-length line and ladder graphs with cyclic or Möbius boundary conditions exhibits a phase transition at finite temperature if 0

Suggested Citation

  • Shrock, Robert, 2000. "Exact Potts model partition functions on ladder graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 283(3), pages 388-446.
    • Handle: RePEc:eee:phsmap:v:283:y:2000:i:3:p:388-446
    DOI: 10.1016/S0378-4371(00)00109-6
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    Cited by:

    1. Chang, Shu-Chiuan & Shrock, Robert, 2023. "Measures of spin ordering in the Potts model with a generalized external magnetic field," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 613(C).
    2. Wu, F.Y & Wang, J, 2001. "Zeroes of the Jones polynomial," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 296(3), pages 483-494.
    3. Chang, Shu-Chiuan & Shrock, Robert, 2020. "Asymptotic behavior of acyclic and cyclic orientations of directed lattice graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    4. Jin, Xian'an & Zhang, Fuji, 2003. "Zeros of the Jones polynomials for families of pretzel links," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 328(3), pages 391-408.

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