Can Lee-Yang zeros theorem account for triple point phase transition?

Question

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?

This post imported from StackExchange Physics at 2023-11-12 18:19 (UTC), posted by SE-user xiaohuamao

Comments

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).

This post imported from StackExchange Physics at 2023-11-12 18:19 (UTC), posted by SE-user Yvan Velenik

---

Note also that if you have three phases, you should consider two external fields.

This post imported from StackExchange Physics at 2023-11-12 18:19 (UTC), posted by SE-user Yvan Velenik

Answer 1

To answer your question, Lee–Yang zeros theorem cannot account for triple point phase transitions in its standard form because the theorem describes binary phase transitions arising from codimension‑1 zero curves in the complex plane of a single external parameter. Triple points require codimension‑2 intersections of zero surfaces in a multi‑parameter partition function.

A consistent extension is possible by promoting the partition function to Z(β,h,p) or Z(β,h,μ), allowing the zero locus to form surfaces whose intersections generate the required degree‑3 singularities. Where pressure p or chemical potential μ acts as a second complex variable. Zeros form surfaces rather than curves, coexistence lines become coexistence sheets, and triple points arise as intersections of zero surfaces.

Comments

It would be nice if you'd give a reference.