Exact correlation functions at finite temperatures in Tomonaga-Luttinger liquid with an open end

Abstract: The paradigmatic state of a 1D collective metal, the Tomonaga-Luttinger liquid (TLL), offers us an exact analytic solution for a strongly interacting quantum system not only for infinite systems at zero temperature but also at finite temperature and with a boundary. Potentially, these results are of high relevance for technology as they could lay the foundation for a many-body description of various nanostructures. For this to happen, we need expressions for local (i.e., spatially resolved) correlations as a function of frequency. In this study, we find such expressions and study their outcome. Based on our analytic expressions we are able to identify two distinct cases of TLL which we call Coulomb metal and Hund metal, respectively. We argue that these two cases span all the situations possible in nanotubes made out of p-block elements. From an applications viewpoint, it is crucial to capture the fact that the end of the 1D system can be coupled with the external environment and emit electrons into it. We discuss such coupling on two levels for both Coulomb and Hund metals: i) in the zeroth order approximation, the coupling modifies the 1D system's boundary conditions; ii) stronger coupling, when the environment can self-consistently modify the 1D system, we introduce spatially dependent TLL parameters. In case ii) we were able to capture the presence of plasmon-polariton particles, thus building a link between TLL and the field of nano-optics.