A Riemann-Hilbert Approach to Slavnov Overlaps in the Lieb-Liniger model

TL;DR

This paper introduces a novel Riemann-Hilbert based method to compute Slavnov overlaps in the Lieb-Liniger model, enabling analysis of quantum many-body overlaps via integral equations and steepest descent techniques.

Contribution

It develops a new approach using Riemann-Hilbert problems to evaluate Slavnov overlaps, expanding analytical tools for integrable models.

Findings

Validated the method by computing the Anderson orthogonality catastrophe at infinite interaction strength.

Demonstrated the applicability of the approach to free fermion limits.

Provided a framework for analyzing overlaps in quantum integrable systems.

Abstract

We provide a method to compute Slavnov overlaps in the Lieb-Liniger model using the steepest descent method of the Riemann-Hilbert problem. To do so, we employ the Matsuo-Kostov Representation of the Slavnov overlaps to write an integral equation for the respective resolvent, and then represent this equation as a Riemann-Hilbert problem. We demonstrate the validity and applicability of the method by computing the Anderson orthogonality catastrophe in the $c\to\infty$ limit, corresponding to free fermions.