Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 337 / 1 / g

The analytic continuation from bosonic imaginary frequency to a retarded frequency is $i{\omega }_n\mapsto \omega +i{0}^{+}$. ThusWriting $E_k=\sqrt{v^2{k}^2+m(T{)}^2}$ and using the Sokhotski–Plemelj formula gives, in this sign convention,The opposite overall convention for the retarded Green function reverses this sign; the corresponding spectral function is conventionally chosen positive at positive frequency.

Near the quantum critical point, $m/T$ is a function only of $\Delta /T$. The Green function has the scaling formwithThis is quantum critical scaling with dynamical critical exponent $z=1$ and leading large-$N$ anomalous dimension $\eta =0$.