Hartz-Millis(HM) theory is a model which exhibits quantum phase transition. The HM action following Altland & Simons is given by $$ S = \frac{1}{\beta}\sum_{\omega_{n}}\int \frac{d^d q}{(2\pi)^d}\left(\delta+q^2+\frac{|\omega_n|}{\Gamma(q)}\right)|\phi|^2+ u\int dx d\tau\ u^4(x,\tau) $$ Where $|\phi|^2 = \phi(q,\omega_{n})\phi(-q,\omega_{-n})$.
Following RG analysis one can find different regimes in the vicinity of the quantum critical point. Of which one particular regime is quantum critical regime. As mentioned by the authors that this region is bounded by the inequality $T \ll r^{z/2}$ and on the other hand if $T \gg r^{z/2}$ corresponds to Fermi liquid regime.
Where $z$ is the dynamical exponent and $$ \delta(b) = b^2 r $$
I understood the reasoning behind the derivation the quantum critical point yet how to derive $T \ll r^{z/2}$ and $T \gg r^{z/2}$ regimes are not clear to me. Can anyone shade some light on this?
