The <analytic continuation> from bosonic imaginary frequency to a retarded frequency is $i\omega_n\mapsto\omega+i0^+$. Thus
$$
\boxed{
G_R(\omega,\mathbf k)
=\frac{vg}{N}
\frac1{v^2k^2+m(T)^2-(\omega+i0^+)^2}.}
$$
Writing $E_k=\sqrt{v^2k^2+m(T)^2}$ and using the <Sokhotski–Plemelj formula> gives, in this sign convention,
$$
\boxed{
\operatorname{Im}G_R(\omega,\mathbf k)
=\frac{\pi vg}{2NE_k}
\left[\delta(\omega-E_k)-\delta(\omega+E_k)\right].}
$$
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 form
$$
G_R(\omega,k;T,\Delta)
=\frac{vg}{N}T^{-2}
\mathcal G\left(\frac\omega T,
\frac{vk}{T},\frac\Delta T\right),
$$
with
$$
\mathcal G^{-1}
=\left(\frac{vk}{T}\right)^2
+\left[2\operatorname{arsinh}
\left(\frac12e^{\Delta/(2T)}\right)\right]^2
-\left(\frac{\omega+i0^+}{T}\right)^2.
$$
This is <quantum critical scaling> with dynamical critical exponent $z=1$ and leading large-$N$ anomalous dimension $\eta=0$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-337/1/g/solution