Statistical mechanics, structure of matter Exactly solved models Proposition

Star–triangle relation

Statement

Let $K_1, K_2, K_3$ be real numbers. Set $c_0 = \cosh(K_1 + K_2 + K_3), \quad c_1 = \cosh(-K_1 + K_2 + K_3), \quad c_2 = \cosh(K_1 - K_2 + K_3), \quad c_3 = \cosh(K_1 + K_2 - K_3).$ Define $L_1 = \tfrac{1}{4} \log \frac{c_0 c_1}{c_2 c_3}, \quad L_2 = \tfrac{1}{4} \log \frac{c_0 c_2}{c_1 c_3}, \quad L_3 = \tfrac{1}{4} \log \frac{c_0 c_3}{c_1 c_2}, \quad R = 2 (c_0 c_1 c_2 c_3)^{1/4}.$ Then for all $\sigma_1, \sigma_2, \sigma_3 \in \set{-1, +1}$ , $\sum_{\sigma_0 \in \set{-1, +1}} e^{\sigma_0 (K_1 \sigma_1 + K_2 \sigma_2 + K_3 \sigma_3)} = R \, e^{L_1 \sigma_2 \sigma_3 + L_2 \sigma_3 \sigma_1 + L_3 \sigma_1 \sigma_2}.$

Remarks

The left side equals $2 \cosh(K_1 \sigma_1 + K_2 \sigma_2 + K_3 \sigma_3)$ , which is unchanged when all three spins flip. So it takes only the four values $2c_0, 2c_1, 2c_2, 2c_3$ , at $(\sigma_1, \sigma_2, \sigma_3) = (+,+,+), (-,+,+), (+,-,+), (+,+,-)$ . The right side has four free constants, and taking logarithms gives a linear system in $\log R, L_1, L_2, L_3$ whose solution is the formula above. In an Ising model, a star is a centre spin joined to three spins with couplings $K_i$ . Summing out the centre replaces the star by a triangle on the three outer spins with couplings $L_i$ . Combined with Kramers–Wannier duality, it relates the Ising models on the honeycomb and triangular lattices to each other and to the square lattice. It is the simplest instance of the Yang–Baxter equation in the form used for spin models.

Going deeper

For a student

The star–triangle transformation lets one move between lattices: decimating every other site of a honeycomb lattice gives a triangular lattice. Baxter turned its generalization into the main engine of exactly solvable models. When the weights depend on a spectral parameter, the star–triangle relation makes transfer matrices with different parameters commute.

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Ising model example
Star–triangle relation statement

Tag	`YC9TY98HJZ`
URI	`m2://YC9TY98HJZ`
Kind	statement, proposition
Writing	W0 stub
MSC	82B20, 82B23
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About	Ising model
Uses	Ising model
Changed	3 November 2026