Past exam of the theoretical physics I course of the University of Cambridge / 2014 / Paper 1 / 6 / b

Take the ordered phase selected by a small positive real field, so $\varphi$ is real and positive before sending $B\to 0^{+}$. Stationarity gives $2a\varphi +2{\varphi }^3=B$. Differentiation yields $\chi =(2a+6{\varphi }^2{)}^{-1}$. ThereforeBoth have mean-field critical exponent one, with a factor-of-two amplitude difference. The below-transition result is the longitudinal response of a selected pure phase. The transverse phase direction is soft, so a transverse zero-field response is singular; an unselected phase average is not the same susceptibility. At $a=0$, $\varphi =(B/2{)}^{1/3}$ and the linear susceptibility diverges.