Finite time path field theory and a new type of universal quantum spin chain quench behaviour

Abstract: We discuss different quench protocols for Ising and XY spin chains in a transverse magnetic field. With a sudden local magnetic field quench as a starting point, we generalize our approach to a large class of local non-sudden quenches. Using finite time path field theory perturbative methods, we show that the difference between the sudden quench and a class of quenches with non-sudden switching on the perturbation vanishes exponentially with time, apart from non-substantial modifications that are systematically accounted for. As the consequence of causality and analytic properties of functions describing the discussed class of quenches, this is true at any order of perturbation expansion and thus for the resummed perturbation series. The only requirements on functions describing the perturbation strength switched on at a finite time t=0 are: (1) their Fourier transform f(p) is a function that is analytic everywhere in the lower complex semiplane, except at the simple pole at p=0 and possibly others with Im (p) < 0, and (2) that f(p)/p converges to zero at infinity in the lower complex semiplane. Prototypical function of this class is tanh(eta t), to which the perturbation strength is proportional after the switching on time t=0. In the limit of large eta, such a perturbation approaches the case of a sudden quench. It is shown that, because of this new type of universal behaviour of Loschmidt echo that emerges in exponentially short time scale, our previous results (Kui\'c, D. et al. Universe 2024, 10, 384) for the sudden local magnetic field quench of Ising and XY chains, obtained by resummation of the perturbative expansion, extend in the long time limit to all non-sudden quench protocols in this class, with non-substantial modifications systematically taken into account. We also show that analogous universal behaviour exists in disorder quenches, and ultimately global ones.