# Gapless topological phases and symmetry-enriched quantum criticality

## TL;DR

This paper develops topological invariants for gapless systems and explores symmetry-enriched quantum criticality, revealing boundary phenomena, edge modes, and classification schemes in 1+1d and higher dimensions.

## Contribution

It introduces a formalism for topological invariants in gapless systems and classifies symmetry-enriched conformal field theories, unifying previous examples and extending to higher dimensions.

## Key findings

- Topological invariants characterize boundary phenomena in gapless systems.
- Edge modes exhibit finite-size splittings depending on gapped sectors.
- Complete classification of symmetry-enriched 1+1d Ising CFTs is provided.

## Abstract

We introduce topological invariants for gapless systems and study the associated boundary phenomena. More generally, the symmetry properties of the low-energy conformal field theory (CFT) provide discrete invariants, establishing the notion of symmetry-enriched quantum criticality. The charges of nonlocal scaling operators, or more generally of symmetry defects, are topological and imply the presence of localized edge modes. We primarily focus on the $1+1d$ case where the edge has a topological degeneracy, whose finite-size splitting can be exponential or algebraic in system size depending on the involvement of additional gapped sectors. An example of the former is given by tuning the spin-1 Heisenberg chain to a symmetry-breaking Ising phase. An example of the latter arises between the gapped Ising and cluster phases: this symmetry-enriched Ising CFT has an edge mode with finite-size splitting $\sim 1/L^{14}$. In addition to such new cases, our formalism unifies various examples previously studied in the literature. Similar to gapped symmetry-protected topological phases, a given CFT can split into several distinct symmetry-enriched CFTs. This raises the question of classification, to which we give a partial answer -- including a complete characterization of symmetry-enriched $1+1d$ Ising CFTs. Non-trivial topological invariants can also be constructed in higher dimensions, which we illustrate for a symmetry-enriched $2+1d$ CFT without gapped sectors.