 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-75/1/c/ii/solution

Expanding

r_{k}

at large positive

g

gives

r_{k} = g - cos k + sin^{2} k / (2 g) + O (g^{- 2})

, so

ϵ_{k} = 2 J g - 2 J cos k + O (J / g) .

The branch is strongly gapped and relatively flat, with width approaching

4 J

against a mean energy of order

2 J g

. These are field-polarized spin-flip quasiparticles with a small relative exchange modulation. Averaging the expansion over momentum also gives

E_{0} / N = - J [g + 1/ (4 g) + O (g^{- 3})]

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