Now the prominent Lee-Yang theorem (or Physical Review 87, 410, 1952) has almost become a standard ingredient of any comprehensive statistical mechanics textbook.

If the volume tends to infinity, some complex roots of the grand canonical partition function may converge to some points $z_0,z_1,z_2,\dots$ on the real axis. Thus these $\{ z_n \}$ divide the complex plane into some isolated phases. According to the singularity near the $\{z_n\}$ every two neighbouring phases may have phase transition phenomena occurring.

Here comes my question. Considering three phases surrounding a triple point in a phase diagram, they can transit to each other (just think about water). Since the neighbourhood along the real axis consists of only two possibilities, I wonder if this theory could account for a description of the triple point. And what is the connection between the neighbourhood of patches on the complex plane and the neighbourhood of phases in a phase diagram?

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    $\begingroup$ If I remember correctly, the paper by Biskup et al, General Theory of Lee-Yang Zeros in Models with First-Order Phase Transitions, arXiv:math-ph/0004003, Phys. Rev. Lett. 84, 4794–4797 (2000), discusses, among others, the Blume-Capel model (which has a triple point). $\endgroup$ Commented Nov 8, 2013 at 6:40
  • $\begingroup$ Note also that if you have three phases, you should consider two external fields. $\endgroup$ Commented Nov 8, 2013 at 7:36