Gapless Topological Phases and Topological Holography

Abstract:

Topological phenomena have traditionally been associated with gapped phases—such as quantum Hall states and topological insulators and superconductors. In recent years, however, it has become clear that topology can also emerge in gapless phases and at critical points. Examples include gapless spin liquids and gapless symmetry-protected topological phases (gSPTs). In this thesis, I develop a general framework for a broad class of gapless topological phases. First, I introduce a cohomological classification of gSPTs based on the topological response theory. I then explore the implications of this classification including lattice model realizations and demonstrating agreement with alternative classification schemes in the literature.

A recent and powerful idea in the study of topological quantum phases is topological holography, which posits that the full topological structure of any phase, gapped or gapless, can be encoded in a topological order in one higher dimension. I show that, for gSPTs, this higher-dimensional bulk provides a complete description: the bulk theory reproduces the cohomological classification and captures all the defining properties of the gSPT. Finally, I offer a brief outlook on gapless topological phases beyond gSPTs, sketching a general theory of all finite-type gapless topological phases and their holographic descriptions.