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

TL;DR

This paper identifies a unique entanglement entropy signature for one-dimensional intrinsically gapless topological phases, showing the absence of Friedel oscillations as a hallmark of their long-range string order.

Contribution

It introduces a novel entanglement entropy diagnostic that distinguishes gapless topological Luttinger liquids from other gapless phases by the absence of Friedel oscillations.

Findings

Entanglement entropy of TLLs lacks Friedel oscillations.

Absence of Friedel oscillations correlates with long-range string order.

Diagnostic is robust and experimentally accessible.

Abstract

Topological quantum matter is typically associated with gapped phases and edge modes protected by the bulk gap. In contrast, recent work (Phys. Rev. B 104, 075132) proposed intrinsically gapless topological phases that, in one dimension, carry protected edge modes only when the bulk is a gapless Luttinger liquid. The edge modes of such a topological Luttinger liquid (TLL) descend from a nonlocal string order that is forbidden in gapped phases and whose precise form depends on the symmetry class of the system. In this work, we propose a powerful and unbiased entanglement-based smoking gun signature of the TLL. In particular, we show that the entanglement entropy profile of a TLL lacks Friedel oscillations that are invariably present in other gapless one dimensional phases such as ordinary Luttinger liquids, and argue that their absence is closely related to a long-ranged string order which is an intrinsic property of the TLL. Crucially, such a diagnostic is more robust against numerical errors and relatively easier to measure in experiments as it relies on the entanglement entropy rather than entanglement spectrum, unlike the entanglement-based diagnostics of gapped topological phases in one dimension.