The main quantity in the study of dynamical quantum phase transition (DQPT) is Loschmidt echo amplitude defined as
$G(t)=\langle \Psi_{0}|\Psi_{0}(t)\rangle=\langle \Psi_{0}|e^{-iHt}|\Psi_{0}\rangle$.
The rate of return probability is given by
$R(t)=-\frac{1}{N}\lim_{N\to \infty}\log[G(t)]$.
The DQPT is signaled by the singular behavior of $R(t)$ at the critical time $t_{c}$.
Replacing the time t by a complex time $z=t+i\tau$, leads to a complex Loschmidt echo amplitude
$G(z)=\langle \Psi_{0}|e^{-zH}|\Psi_{0}\rangle$.
The zeros of the function $G(z)$ are called Fisher zeros lying on the complex time ($z$) plane. The Fisher zeros form a structure and when they cross the real time axis, the DQPT occurs at real time $t_{c}$.
As it is not always the case to find analytical formula for the Fisher-zeros, I am looking for methods to calculate the Fisher zeros numerically. I consider that the Hamiltonian can be represented as a finite dimensional matrix. A physical system could be a system of non-interacting fermions in one spatial dimension. $|\Psi_{0}\rangle$ is also accessible numerically and $G(t)$ can be numerically computed for large system size.
To find your answer, please look at this paper:
Article Dynamical quantum phase transitions and the Loschmidt echo: ...
You can either find them by looking for the simultaneous conditions via a graphical method (rough estimation of the zero location) or minimizing if you want numerical. Use the light cone renormalization group (LCRG) algorithm from T. Enss and J. Sirker, New J. Phys.14, 023008 (2012). Example: https://arxiv.org/pdf/1312.4165.pdf
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