Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 344 / 1 / d

Solution

Choose the translationally invariant Gaussian variational approximation

H_{0} = \sum_{q}^{+} J (q) ∣ ϕ_{q} ∣^{2}, J (q) > 0.

The superscript

+

means that one member of each pair

{q, - q}

is included; this is necessary because a real field obeys

ϕ_{- q} = ϕ_{q}^{*}

. The constrained zero mode is omitted.

For each independent complex mode,

\int_{R^{2}} d (Re ϕ_{q}) d (Im ϕ_{q}) e^{- J (q) ∣ ϕ_{q} ∣^{2}} = \frac{π}{J ( q )} .

Consequently,

F_{0} = \sum_{q}^{+} lo g \frac{J ( q )}{π}, ⟨ H_{0} ⟩_{0} = \sum_{q}^{+} 1, ⟨ \sum_{q}^{+} G (q) ∣ ϕ_{q} ∣^{2} ⟩_{0} = \sum_{q}^{+} \frac{G ( q )}{J ( q )} .

F = \sum_{q}^{+} [lo g \frac{J ( q )}{π} - 1 + \frac{G ( q )}{J ( q )}] + g \int d^{d} r ⟨ ∣ ϕ (r) ∣^{3} ⟩_{0},

which is the required upper bound.

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