 Solution

ID: past-exam-of-the-advanced-quantum-theory-path-integrals-and-many-particle-physics-course-of-the-university-of-oxford/2025/1/a/i/solution

Each neighboring pair of Ising spins contributes at least

- J

to the energy, with equality exactly when the two Ising spins are opposite. For even

N

, alternation closes consistently under the periodic boundary conditions. The ground states are therefore

S_{j} = σ (- 1)^{j - 1}, σ \in {+ 1, - 1}, E_{0} = - J N, g_{0} = 2.

Here

g_{0}

S_{1}

fixes every other Ising spin, so these are all the ground states; translations do not give additional spin configurations. The two ground states are exchanged by spin inversion symmetry.

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