# How to numerically calculate Fisher zeros?

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.

• I doubt you can do it in a general case, it will depend on how the Hamiltonian looks like and whether the Hilbert space is finite or infinite-dimensional, I guess. – Bzazz Sep 5 '18 at 8:30