Statistical mechanics, structure of matter Lattice spin systems Example

Ashkin–Teller model

Example

Let $G = (V, E)$ be a finite graph, let $J_1, J_2, J_4$ be real constants and let $\beta \ge 0$ be the inverse temperature. A configuration assigns to each vertex $x$ two Ising spins $\sigma_x, \tau_x \in \set{-1, +1}$ . Its energy is $H(\sigma, \tau) = -\sum_{\set{x, y} \in E} \left( J_1 \sigma_x \sigma_y + J_2 \tau_x \tau_y + J_4 \sigma_x \sigma_y \tau_x \tau_y \right).$ The Ashkin–Teller model is the probability measure on $(\set{-1, +1}^2)^V$ proportional to $e^{-\beta H(\sigma, \tau)}$ .

Remarks

If $J_4 = 0$ , the model is two independent Ising models, one for $\sigma$ and one for $\tau$ . If $J_1 = J_2 = J_4 = J$ , the bond energy is $-3J$ when $(\sigma_x, \tau_x) = (\sigma_y, \tau_y)$ and $+J$ otherwise. So it equals $J - 4J \delta_{(\sigma_x, \tau_x), (\sigma_y, \tau_y)}$ , and the model is the four-state Potts model with coupling $4J$ . Ashkin and Teller introduced it in 1943 as a model of a four-component system. On the square lattice it has a line of critical points along which the critical exponents vary continuously.

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Graph concept
Ising model example
Lattice model concept
Potts model example
Ashkin–Teller model example

Tag	`BYZFYS96M7`
URI	`m2://BYZFYS96M7`
Kind	example
Writing	W0 stub
MSC	82B20, 82B23
Instance of	Lattice model
Uses	Ising model, Potts model
Changed	3 November 2026

Ashkin–Teller model

Example

Remarks

Reading path

Further reading

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