Past exam of the theoretical physics major option course of the University of Oxford / 2018 / 6 / d

Write $a=a_0(T-T_c)$, $a_0>0$, at fixed $b\ge 0$, and define the critical exponent by $|m|\sim (T_c-T{)}^{\beta }$ on the ordered side. For fixed $b>0$ and small negative $a$, the stationary equation gives $m^2=-a/b+O(a^2)$. ConsequentlyAt the tricritical point approached along the tuned path $b=0$, the stationary equation instead gives $m^4=-a/c$, soFor fixed $b<0$, the transition occurs before $a$ reaches zero and the magnetization jumps; it is not a second-order transition and has no vanishing-order-parameter exponent of this form. A path that does not keep the quartic coefficient tuned to zero generally crosses over to the ordinary result.