  The weakly interacting helical Luttinger liquid, due to spin momentum
locking, is characterized by extremely peculiar local observables: we show that
the density-density correlation functions do not exhibit signatures of Friedel
and Wigner oscillations, and that spin-spin correlation functions, which are
strongly anisotropic, witness the formation of a planar spin wave. Moreover, we
demonstrate that the most relevant scattering potentials involving a localized
impurity are not able to modify the electron density, while only magnetic
impurities can pin the planar spin density wave.
