An Intrinsically Gapless Topological Phase with Dipole Symmetry

Abstract

We show that the boundary fractionalization familiar from gapped topological insulators can enforce bulk gaplessness in a dipole-conserving system. In a one-dimensional fermion model with charge conservation, dipole conservation, and particle-hole symmetry, we realize an intrinsically gapless symmetry-protected topological (igSPT) phase combining a dipolar Luttinger liquid with half-quantized boundary charges. This boundary structure obstructs a unique gapped bulk preserving these symmetries and lattice translation. Remarkably, the two charge-neutral edge configurations remain exactly degenerate even at finite size due to dipole conservation and particle-hole symmetry. The boundary fractionalization also manifests as shifted conformal towers in the dipole-resolved entanglement spectrum, reflecting distinct conformal boundary conditions of the Luttinger liquid. We support our analysis of the igSPT by studying exactly-solvable limits and performing DMRG computations, establishing its stability for a wide range of tuning parameters.