# Free fermions at the edge of interacting systems

## TL;DR

This paper investigates how interactions affect the edge behavior of inhomogeneous one-dimensional quantum systems, showing that Tracy-Widom universality persists due to vanishing density at the edge, with numerical validation and exploration of new universality classes.

## Contribution

It demonstrates that strong bulk interactions are renormalized to zero at the edge, preserving Tracy-Widom scaling, and provides exact length scale predictions for integrable systems, supported by numerical results.

## Key findings

- Tracy-Widom distribution remains at the edge despite interactions.
- Numerical agreement with analytical predictions of length scale.
- Identification of new universality classes at quantum quench fronts.

## Abstract

We study the edge behavior of inhomogeneous one-dimensional quantum systems, such as Lieb-Liniger models in traps or spin chains in spatially varying magnetic fields. For free systems these fall into several universality classes, the most generic one being governed by the Tracy-Widom distribution. We investigate in this paper the effect of interactions. Using semiclassical arguments, we show that since the density vanishes to leading order, the strong interactions in the bulk are renormalized to zero at the edge, which simply explains the survival of Tracy-Widom scaling in general. For integrable systems, it is possible to push this argument further, and determine exactly the remaining length scale which controls the variance of the edge distribution. This analytical prediction is checked numerically, with excellent agreement. We also study numerically the edge scaling at fronts generated by quantum quenches, which provide new universality classes awaiting theoretical explanation.