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

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