Absence of Friedel oscillations in the entanglement entropy profile of one-dimensional intrinsically gapless topological phases

Topological quantum matter is typically associated with gapped phases. For instance, the bulk energy gap of symmetry-protected topological phases localizes edge modes near the boundary. Here, we study the proposed intrinsically gapless topological phase (Phys. Rev. B 104, 075132) in one dimension -- or topological Luttinger liquid -- via DMRG. For such phases, the string order forbidden in gapped phases defines the bulk-edge correspondence and protects the edge modes. In this work, we examine the entanglement entropy profile of the phase and find that it lacks Friedel oscillations that ordinary Luttinger liquids invariably contain. We find that the lack of Friedel oscillations stems from the mere existence of a string order regardless of the precise form of the order parameter. Thus, it gives us a more unbiased way to diagnose the above topological phase. It would be interesting to study other gapless topological phases using this method. 

 This work has been archived as arXiv:2201.07260