Phase transition in complex-time Loschmidt echo of short and long range spin chain

We explain and exploit the random matrix formulation of the Loschmidt echo for the XX spin chain, valid for multiple domain wall initial states and also for a XX spin chain generalized with additional interactions to more neighbours. For models with interactions decaying as $e^{-\alpha \left\vert l-j\right\vert }/\left\vert l-j\right\vert ^{p+1}$, with $p$ integer or natural number and $\alpha \geq 0$, we show that there are third order phase transitions in a double scaling limit of the complex-time Loschmidt echo amplitudes. For the long-range version of the chain, we use an exact result for Toeplitz determinants with a pure Fisher-Hartwig singularity, to obtain exactly the Loschmidt echo for complex times and discuss the associated Stokes phenomena. We also study the case of a finite chain for one of the generalized XX models.