Correlation functions for the detection of Wigner molecules in a one-channel Luttinger liquid quantum dot

In one-channel, finite-size Luttinger one-dimensional quantum dots, both Friedel oscillations and Wigner correlations induce oscillations in the electron density with the same wavelength, pinned at the same position. Therefore, observing such a property does not provide any hint about the formation of a Wigner molecule when electrons interact strongly and other tools must be employed to assess the formation of such correlated states. We compare here the behavior of three different correlation functions and demonstrate that the integrated two point correlation function, which represents the probability density of finding two particles at a given distance, is the only faithful estimator for the formation of a correlated Wigner molecule.