Dynamical response functions in the quantum Ising chain with a boundary

TL;DR

This paper calculates dynamical response functions in the quantum Ising chain with a boundary, revealing oscillatory behaviors and boundary effects using form factor expansions in the scaling limit.

Contribution

It introduces a spectral form factor expansion for boundary response functions in the quantum Ising chain, accounting for boundary bound states and late-time oscillations.

Findings

Correlators exhibit rapid convergence for certain spatial ratios.

Oscillatory behavior observed at late times far from the boundary.

Boundary bound states influence the dynamical response.

Abstract

We determine dynamical response functions $<{\cal O}^\dagger(t,x_1){\cal O}(0,x_2)>$ in the scaling limit of the quantum Ising chain on the half line in the presence of a boundary magnetic field. Using a spectral representation in terms of infinite volume form factors and a boundary state, we derive an expansion for the correlator that is found to be rapidly convergent as long as $|\frac{x_1+x_2}{\xi}|\agt 0.2$ where $\xi$ is the correlation length. At sufficiently late times we observe oscillatory behaviour of the correlations arbitrarily far away from the boundary. We investigate the effects of the boundary bound state that is present for a range of boundary magnetic fields.