# Inhomogeneous quenches in the transverse field Ising chain: scaling and front dynamics

## TL;DR

This paper analytically studies the non-equilibrium dynamics of the transverse field Ising chain after an inhomogeneous quench, revealing detailed front structures, hydrodynamic behavior, and lattice effects.

## Contribution

It provides exact integral formulas for correlation functions and analyzes front dynamics, including the Airy kernel description and energy back-flow phenomena.

## Key findings

- Correlation functions expressed as integrals for all times and fields
- Hydrodynamic profiles derived from quasiparticle picture
- Front regions described by Airy kernel and derivatives

## Abstract

We investigate the non-equilibrium dynamics of the transverse field quantum Ising chain evolving from an inhomogeneous initial state given by joining two macroscopically different semi-infinite chains. We obtain integral expressions for all two-point correlation functions of the Jordan-Wigner Majorana fermions at any time and for any value of the transverse field. Using this result, we compute analytically the profiles of various physical observables in the space-time scaling limit and show that they can be obtained from a hydrodynamic picture based on ballistically propagating quasiparticles. Going beyond the hydrodynamic limit, we analyze the approach to the non-equilibrium steady state and find that the leading late time corrections display a lattice effect. We also study the fine structure of the propagating fronts which are found to be described by the Airy kernel and its derivatives. Near the front we observe the phenomenon of energy back-flow where the energy locally flows from the colder to the hotter region.