 Solution

ID: past-exam-of-the-quantum-condensed-matter-field-theory-course-of-the-university-of-cambridge/2014/paper-1/1/d/solution

The inverse transformation has negative off-diagonal entries. Choosing

tanh 2 θ_{k} = γ_{k}

removes anomalous terms and gives

ω_{k} = 2 J S 1 - γ_{k}^{2} = 2 J S ∣ sin k ∣, H = - J NS (S + 1) + k \sum ω_{k} (α_{k}^{†} α_{k} + 1/2) + O (S^{0}) .

There are gapless points at

k = 0

and the zone boundary; near zero,

ω_{k} ≃ 2 J S ∣ k ∣

2 J S

in the stated units. The zero modes are collective rotations: their divergent squeezing cannot be treated as ordinary normalized independent oscillator vacua. A finite-size cutoff or infinitesimal symmetry-breaking field regulates them.

- J S (S + 1) + (2/ π) J S

. This is a spin-wave prediction, not an exact spectrum for every finite-

S

one-dimensional chain.

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