The main candidate for the superfluid pathways in solid Helium-4 are dislocations with Burgers vector along the hcp symmetry axis. A generic edge dislocation which can perform superclimb — climb supported by the superflow along its core – has been found by Monte Carlo simulations to persist in several quantum phases: i) Tomonago-Luttinger Liquid (TLL}, where the superclimb is suppressed while superfluidity along the core persists; ii) The insulator which is non-superclimbing and non- superfluid ; iii) non-TLL which is superclimbing phase characterized by parabolic excitation spectrum of supercurrents. The phase (ii) occurs regardless of the filling factor with respect to the hosting lattice. The phase (iii) is characterized by ODLRO which, however, is fragile at any finite temperature – resulting in the exponential decay of the density matrix. The transition TLL- insulator is shown to belong to the Berizinskii-Kosterlitz- Thouless universality despite that the elementary scaling analysis does not predict such a transition. The transformation TLL to non-TLL or insulator to non-TLL are induced by bias by chemical potential, and it exhibits strong hysteresis consistent with the Ist order transition. It is argued that, in spite of the “no-go” theorem for a transition at finite T in 1D, the transition may persist at finite temperature. It is also shown that the long-range interactions due to elastic forces in the crystal bulk do not change this picture qualitatively.
