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The notion of a crosscap state, a special conformal boundary state firstdefined in 2d CFT, was recently generalized to 2d massive integrable quantumfield theories and integrable spin chains. It has been shown that the crosscapstates preserve integrability. In this work, we first generalize this notion tothe Lieb-Liniger model, which is a prototype of integrable non-relativisticmany-body systems. We then show that the defined crosscap state preservesintegrability. We derive the exact overlap formula of the crosscap state andthe on-shell Bethe states. As a byproduct, we prove the conjectured overlapformula for integrable spin chains rigorously by coordinate Bethe ansatz. Itturns out that the overlap formula for both models take the same form as aratio of Gaudin-like determinants with a trivial prefactor. Finally we studyquench dynamics of the crosscap state, which turns out to be surprisinglysimple. The stationary density distribution is simply a constant. We alsoderive the analytic formula for dynamical correlation functions in theTonks-Girardeau limit.

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