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Aiming at studying the emergence of Non-Equilibrium Steady States (NESS) inquantum integrable models by means of an exact analytical method, we focus onthe Tonks-Girardeau or hard-core boson limit of the Lieb-Liniger model. Weconsider the abrupt expansion of a gas from one half to the entire confiningbox, a prototypical case of inhomogeneous quench, also known as "geometricquench". Based on the exact calculation of quench overlaps, we develop ananalytical method for the derivation of the NESS by rigorously treating thethermodynamic and large time and distance limit. Our method is based on complexanalysis tools for the derivation of the asymptotics of the many-bodywavefunction, does not make essential use of the effectively non-interactingcharacter of the hard-core boson gas and is sufficiently robust forgeneralisation to the genuinely interacting case.

Non-Equilibrium Steady State of the Lieb-Liniger model: exact treatment of the Tonks Girardeau limit