Gapless topological behaviors in a long-range quantum spin chain

Topology in condensed matter physics is typically associated with a bulk energy gap. However, recent research has shifted focus to quantum critical points or phases that exhibit nontrivial topological properties. In this work, we explore a cluster Ising chain with long-range antiferromagnetic interactions that decay as a power law with distance. Using large-scale density matrix renormalization group simulations, we unambiguously demonstrate that the nontrivial topology at the critical point remains stable against long-range interactions, resulting in a topologically nontrivial critical line. Remarkably, even within the gapped region, the interplay between topology and long-range interaction can give rise to a new topological phase featuring algebraic decay correlation and edge modes, similar to gapless topological phases. Hence, we refer to this new phase as the algebraic topological phase, which exhibits nontrivial gapless topological behaviors and arises solely from long-range interactions without any short-range counterpart.