## SciPost Submission Page

# Free fermions at the edge of interacting systems

### by Jean-Marie Stéphan

#### - Published as SciPost Phys. 6, 057 (2019)

### Submission summary

As Contributors: | Jean-Marie Stéphan |

Arxiv Link: | https://arxiv.org/abs/1901.02770v3 |

Date accepted: | 2019-05-01 |

Date submitted: | 2019-04-26 |

Submitted by: | Stéphan, Jean-Marie |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Quantum Physics |

Approaches: | Theoretical, Computational |

### 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.

### Ontology / Topics

See full Ontology or Topics database.Published as SciPost Phys. 6, 057 (2019)

### Author comments upon resubmission

I am happy to submit a new version, with various small improvements.

### List of changes

I made the following changes in response to the remarks by the second referee.

1) and 2) This is correct, these typos are now corrected. Regarding the question, I believe the corrections are no greater than those for free fermions.

3) I am grateful to the referee for pointing out this reference, which I had missed. This work is now explicitly mentioned in the text.

4) I tried to improve this subsection, as well as several other related points in the manuscript. The discussion is hopefully easier to follow now. The referee is of course right that the Airy kernel accounts for an infinite number of particles, not one as previously stated. This is corrected now; what I meant to point out was that the T-W distribution itself accounts for only one particle, in references to other claims such as those of Ref. 55.

5) Those are now fixed.

I also made other minor changes not requested by the referees.