Absence of orthogonality catastrophe after a spatially inhomogeneous interaction quench in Luttinger liquids

TL;DR

This paper demonstrates that a spatially inhomogeneous interaction quench in Luttinger liquids does not cause an orthogonality catastrophe, with the Loschmidt echo remaining finite and steady state overlaps being analytically derived and numerically validated.

Contribution

It provides an analytic solution for the Loschmidt echo after an inhomogeneous quench in Luttinger liquids and shows the absence of orthogonality catastrophe, validated by numerical simulations.

Findings

No orthogonality catastrophe occurs after inhomogeneous interaction quench.

The steady state Loschmidt echo equals the square of the adiabatic overlaps.

Analytic solutions are validated by numerical simulations on the XXZ chain.

Abstract

We investigate the Loschmidt echo, the overlap of the initial and final wavefunctions of Luttinger liquids after a spatially inhomogeneous interaction quench. In studying the Luttinger model, we obtain an analytic solution of the bosonic Bogoliubov-de Gennes equations after quenching the interactions within a finite spatial region. As opposed to the power law temporal decay following a potential quench, the interaction quench in the Luttinger model leads to a finite, hardly time dependent overlap, therefore no orthogonality catastrophe occurs. The steady state value of the Loschmidt echo after a sudden inhomogeneous quench is the square of the respective adiabatic overlaps. Our results are checked and validated numerically on the XXZ Heisenberg chain.