 Solution

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

For the classical XY ferromagnet choose every spin parallel within the

x y

plane. Each nearest-neighbor and next-nearest-neighbor in-plane dot product is

S^{2}

. The printed sum counts one bond of each range per spin, so

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

There is no additional factor of two from counting both directions: the Hamiltonian operator already lists only

n \to n + 1

and

n \to n + 2

. Alignment along

z

would give zero XY exchange energy and is not the in-plane ordered state used for these expressions.

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