In a typical quantum quench protocol, a Hamiltonian is considered of the form $$H = H_{0} + g H_{1}$$ where $g>g_{c}$, system will be in one phase and $g<g_{c}$, system will be in another phase. This is true for any system that undergoes phase transition. Suppose I prepare the initial state of the system in ground state of $g = g_{1}<g_{c}$ Hamiltonian and at time $t=t_{0}$, I suddenly quench the parameter $g=g_{2}>g_{c}$ and evolve the system, the system at time $t>t_{0}$ is
$$\vert \psi (t) \rangle = e^{-iH(g_{2})t} \vert \psi_{0}(g_{1})\rangle$$.
Then one can ask the probability of how much of the initial state is still available in the state at time $t$. This is given by the Loschmidt echo defined as
$$\mathcal{L}(t)=\vert\langle \psi(g_{1})\vert e^{-iH(g_{2})t}\vert \psi(g_{1})\rangle\vert^2$$. $$\mathcal{L}(t)= \sum_{n}\vert e^{-iE_{n}(g_{2})t}\vert\langle\psi_{n}(g_{2})\vert\psi_{0}(g_{1})\rangle\vert^2 \vert^2$$ Time averaging of Loschmidt echo in the long time limit $t\to\infty$, gives us the Inverse participation ratio
$$\bar{\mathcal{L}} = \sum_{n}\vert \langle\psi_{n}(g_{2})\vert\psi_{0}(g_{1})\rangle \vert^4 = IPR $$
How is this IPR can be used as a signature of quantum phase transition? I mean I start with $g<g_{c}$ and quench to new $g>g_{c}$, how this quantity helps us to know that I crossed phase transition critical point $g_{c}$?
p.s: I am not interested in dynamical quantum phase transition, I am interested more about general quantum phase transition. not PTs in dynamical sense. Any suggestion is appreciated.
