Topology and edge modes in quantum critical chains

TL;DR

This paper demonstrates that topology can protect zero energy edge modes at critical points in 1D quantum chains, even without bulk gaps, and provides a classification of such phases using topological invariants and conformal field theory.

Contribution

It introduces a topological invariant for critical phases in non-interacting BDI class chains and classifies these phases by central charge and invariant, extending understanding of edge modes at criticality.

Findings

Topological edge modes can exist at critical points without bulk gaps.

A topological invariant classifies critical phases in non-interacting BDI chains.

Edge modes remain stable under interactions and disorder.

Abstract

We show that topology can protect exponentially localized, zero energy edge modes at critical points between one-dimensional symmetry protected topological phases. This is possible even without gapped degrees of freedom in the bulk ---in contrast to recent work on edge modes in gapless chains. We present an intuitive picture for the existence of these edge modes in the case of non-interacting spinless fermions with time reversal symmetry (BDI class of the tenfold way). The stability of this phenomenon relies on a topological invariant defined in terms of a complex function, counting its zeros and poles inside the unit circle. This invariant can prevent two models described by the \emph{same} conformal field theory (CFT) from being smoothly connected. A full classification of critical phases in the non-interacting BDI class is obtained: each phase is labeled by the central charge of the CFT, $c \in \frac{1}{2}\mathbb N$, and the topological invariant, $\omega \in \mathbb Z$. Moreover, $c$ is determined by the difference in the number of edge modes between the phases neighboring the transition. Numerical simulations show that the topological edge modes of critical chains can be stable in the presence of interactions and disorder.