Quantum phase transition occur where $T=0$.
$$ \hat{H} = \hat{A} + g\hat{B} \\ [\hat{A},\hat{B}]\neq 0 $$
where $g \ll g_c$
$$ \hat{H} \sim \hat{A} $$
$$ \left|\psi_g\right> = \left|\psi(\hat{A})\right> $$
where $g \gg g_c$
$$ \hat{H} \propto \hat{B} $$
$$ \left|\psi_g\right> = \left|\psi(\hat{B})\right> $$
Quantum fluctuation is significant where $g\sim g_c$
The dimensionless distance from a critical point
$$ \epsilon = \frac{g-g_c}{g_c}=\frac{g}{g_c}-1 $$
gap size
$$ \Delta_{\rm gap} = 2 |g-g_c| = 2g_c|\epsilon| $$
Relaxation time