The critical <two-point correlation function> shows that the field has scaling dimension
$$
\Delta_\phi=\frac{d-2+\eta}{2}.
$$
In the ordered phase the <correlation length> is the only diverging scale, so the <order parameter> scales as
$$
\phi\sim\xi^{-\Delta_\phi}
\sim t^{\nu\Delta_\phi}.
$$
Therefore
$$
\boxed{\beta=\frac\nu2(d-2+\eta).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 303 / 3 / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-303/3/ii/solution