Integrable Crosscap States: From Spin Chains to 1D Bose Gas

TL;DR

This paper extends the concept of crosscap states to the Lieb-Liniger model, demonstrating their integrability preservation, deriving exact overlaps, and analyzing quench dynamics and correlation functions in 1D Bose gases.

Contribution

It generalizes crosscap states to the Lieb-Liniger model, proves overlap formulas rigorously, and explores their dynamical properties in integrable 1D Bose gases.

Findings

Crosscap states preserve integrability in the Lieb-Liniger model.

Exact overlap formulas are derived as ratios of Gaudin-like determinants.

Quench dynamics of crosscap states show a simple stationary density distribution.

Abstract

The notion of a crosscap state, a special conformal boundary state first defined in 2d CFT, was recently generalized to 2d massive integrable quantum field theories and integrable spin chains. It has been shown that the crosscap states preserve integrability. In this work, we first generalize this notion to the Lieb-Liniger model, which is a prototype of integrable non-relativistic many-body systems. We then show that the defined crosscap state preserves integrability. We derive the exact overlap formula of the crosscap state and the on-shell Bethe states. As a byproduct, we prove the conjectured overlap formula for integrable spin chains rigorously by coordinate Bethe ansatz. It turns out that the overlap formula for both models take the same form as a ratio of Gaudin-like determinants with a trivial prefactor. Finally we study quench dynamics of the crosscap state, which turns out to be surprisingly simple. The stationary density distribution is simply a constant. We also derive the analytic formula for dynamical correlation functions in the Tonks-Girardeau limit.