Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 337 / 1 / e

Subtract the zero-temperature term by using $\coth (x/2)=1+2/(e^x-1)$. In three dimensions the thermal part iswhere the last integral uses the Bose-Einstein distribution. Gaussian phase fluctuations reduce the order parameter by $⟨e^{-i\theta }⟩=e^{-⟨{\theta }^2⟩/2}$. Taking symmetry restoration to occur when the thermal variance is of order one givesThe effective action in part c has $\chi =1/(2\lambda )$, and hence $T_{\ast }\sim \sqrt{6v^3/\lambda }$. This is an order-of-magnitude estimate because near restoration the phase-only effective field theory omits large amplitude fluctuations and sensitivity to its ultraviolet cutoff.