Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 303 / 1 / b / i

Table of contents
- Solution i

Solution

Write

σ_{i} = m + δ σ_{i}

and neglect products of two fluctuations. Then

σ_{i} \cdot σ_{j} ≃ m \cdot (σ_{i} + σ_{j}) - m^{2} .

Each site has

q

neighbours and each bond is counted once, so the mean-field approximation gives

E_{MF} = - J q m \cdot \sum_{i} σ_{i} + \frac{N J q}{2} m^{2} .

Choose the

x

axis along

m

. For the four-state clock model, the single-site partition function is

z_{1} = \sum_{θ = 0, π /2, π, 3 π /2} e^{β J q m c o s θ} = 2 + 2 cosh (β J q m) .

Consequently

f = \frac{J q}{2} m^{2} - T lo g [2 + 2 cosh (β J q m)],

and therefore

A = \frac{J q}{2}, C = 2, D = 2 .

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