Edge States at Phase Boundaries and Their Stability

TL;DR

This paper investigates how Robin boundary conditions induce edge states in quantum field theories, analyzing their stability, bounds, and implications for topological phases like quantum Hall effects and insulators.

Contribution

It provides a rigorous analysis of edge state existence, stability bounds, and their dependence on boundary conditions and system size in various quantum field theories.

Findings

Edge states always appear under Robin boundary conditions.

Spectral lower bounds ensure stability of massive theories.

Edge states can disappear in small systems, ensuring stability.

Abstract

We analyse the effects of Robin boundary conditions on quantum field theories of spin 0, 1 and 1/2. In particular, we show that these conditions always lead to the appearance of edge states that play a significant role in quantum Hall effect and topological insulators. We prove in a rigorous way the existence of spectral lower bounds on the kinetic term of the Hamiltonian, which guarantees the stability and consistency of massive field theories when the mass is larger than the lower bound of the kinetic term. We also find an upper bound for the deepest edge state. The explicit dependence of both bounds on the boundary conditions and the size of the system is derived under very general conditions. For fermionic systems we analyse the case of Atiyah-Patodi-Singer and chiral bag boundary conditions. We point out the existence of edge states also in these cases and show that they disappear for small enough systems. Stability is granted in this case.