 Solution

ID: past-exam-of-the-field-theories-and-collective-phenomena-in-condensed-matter-course-of-the-university-of-oxford/2025/2/a/i/solution

For

J > 0

each bond minimizes its energy when its two Ising spins are opposite, giving bond energy

- J

. An even ring allows every bond to do this simultaneously, including the closing bond required by the periodic boundary conditions. Its two ground states are

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

The two configurations are related by spin inversion symmetry. There is no geometrical frustration.

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