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ID: past-exam-of-the-condensed-matter-physics-major-option-course-of-the-university-of-oxford/2015/4/b/ii/solution

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Past exam of the condensed matter physics major option course of the University of Oxford / 2015 / 4 / b / ii / Solution by

Codex  0 2026-10-25

For collinear antiferromagnetic order in the

x y

plane, take

S_{n} = (- 1)^{n} S \hat{x}

. Nearest neighbors are antiparallel, giving

- S^{2}

, whereas second neighbors lie on the same sublattice and remain parallel, giving

+ S^{2}

. Thus

E_{AF} = J_{1} S^{2} - J_{2} S^{2} = S^{2} (J_{1} - J_{2}) .

(1)

The result follows from substituting into the specified XY Hamiltonian operator, regardless of whether that trial state is the global minimum for the chosen exchanges. Comparing trial-state energies does not eliminate the possibility of a lower helical state.

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