Failure of the Quench Action Formalism for Mott Insulator Initial States

Abstract

The quench action formalism relies on the assumption that the overlap between a generic initial state $\left|\Psi_{0}\right\rangle $ and an eigenstate of an integrable model - defined through the rapidities $\left|k_{1},...k_{N}\right\rangle $ - can be written as: \begin{equation} \left\langle k_{1},...k_{N}\mid\Psi_{0}\right\rangle =\exp\left(-S_{\Psi_{0}}\left(\rho\left(k\right)\right)\right),\label{eq:Exponential} \end{equation} where $\rho\left(k\right)$ is the quasiparticle density of the state $\left|k_{1},...k_{N}\right\rangle $ and $S_{\Psi_{0}}$ is some smooth function of $\rho\left(k\right)$ that depends on $\Psi_{0}$. In particular the quench action formalism assumes the overlap depends smoothly on the quasiparticle density $\rho\left(k\right)$. In this work, by explicit counter example, we show that this is not the case. We consider the quench between a Mott insulator and a Lieb Liniger gas. We show that the overlap between the ground state of the Mott insulator and arbitrary eigenstates of the Lieb Liniger gas has a highly singular behavior and no expression like Eq. (1) applies. We do so within the Tonks Girardeau limit of the Lieb Liniger gas and to leading order in the $1/c$ expansion for the overlap (with $c$ being the coupling constant of the Lieb Liniger gas). In the Appendix we show similar results for overlaps in the XXZ model with crystal states.