# Return amplitude after a quantum quench in the XY chain

## TL;DR

This paper derives an exact formula for transition amplitudes in the quantum XY chain, analyzes return amplitudes for specific states, and explores finite-size effects and dynamical phase transitions.

## Contribution

It provides a new exact analytical expression for transition amplitudes and investigates finite-size effects and dynamical phase transitions in the XY chain.

## Key findings

- Exact formula for transition amplitudes between eigenstates.
- Analytical expression for return amplitude of polarized and Nél states.
- Finite-size effects and traversal phenomena depend on initial state and system size.

## Abstract

We determine an exact formula for the transition amplitude between any two arbitrary eigenstates of the local $z$-magnetization operators in the quantum XY chain. We further use this formula to obtain an analytical expression for the return amplitude of fully polarized states and the N\'eel state on a ring of length $L$. Then, we investigate finite-size effects in the return amplitude: in particular quasi-particle interference halfway along the ring, a phenomenon that has been dubbed traversal~\cite{FE2016}. We show that the traversal time and the features of the return amplitude at the traversal time depend on the initial state and on the parity of $L$. Finally, we briefly discuss non-analyticities in time of the decay rates in the thermodynamic limit $L\rightarrow\infty$, which are known as dynamical phase transitions.