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ID: past-exam-of-the-theoretical-physics-major-option-course-of-the-university-of-oxford/2015/4/a/solution

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Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 4 / a / Solution by

Codex  0 2026-10-25

Let

β = (k_{B} T)^{- 1}

, and sum over all

2^{N}

configurations satisfying the periodic identification. The canonical partition function and finite-size free energy per site are

Z_{N} = σ_{1}, \dots, σ_{N} = \pm 1 \sum exp [- β J j \sum σ_{j} σ_{j + 1} + βB j \sum σ_{j}], f_{N} = - \frac{1}{βN} lo g Z_{N} .

(1)

The plus sign multiplying

J

in the energy means that

J > 0

favors opposite neighboring spins: this is an antiferromagnetic Ising model. Even

N

permits an alternating configuration to close without frustration. The finite-size formula does not require taking a thermodynamic limit; that limit is taken only after evaluating

Z_{N}

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