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

TL;DR

This paper introduces a universal behavior in quantum spin chains after various local and global quenches, showing that non-sudden quenches rapidly resemble sudden quenches in their long-time dynamics, using finite time path field theory.

Contribution

The study extends previous sudden quench results to a broad class of non-sudden quenches, demonstrating exponential convergence and universality in quantum spin chain dynamics.

Findings

Universal exponential decay of differences between quenches

Long-time behavior of Loschmidt echo is similar across quench types

Applicability to disorder and global quenches

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 (FTPFT) 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 as follows: (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) $f(p)/p$ converges to zero at infinity in the lower complex semiplane. A prototypical function of this class is $\tanh(\eta t)$, to which the perturbation strength is proportional after the switching at 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 behavior of Loschmidt echo (LE) that emerges in an exponentially short time scale, our previous results for the sudden local magnetic field quench of Ising and XY chains, obtained by the 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 behavior exists in disorder quenches, and ultimately global ones.