Massless Fermions on a half-space: The curious case of 2+1-dimensions

TL;DR

This paper explores boundary conditions for massless Dirac fermions in 2+1 dimensions on a half-plane, revealing special cases that preserve certain symmetries and lead to edge states with exotic properties and ferromagnetism.

Contribution

It identifies unique boundary conditions that preserve CPT and Poincare symmetries, and analyzes the resulting edge states and their exotic symmetry representations.

Findings

Single fermion species boundary condition preserves CPT and full symmetries.

Doubled fermions boundary condition admits fermion zero modes.

Edge states induce exotic symmetry representations and ferromagnetism.

Abstract

Boundary conditions for a massless Dirac fermion in 2+1 dimensions where the space is a half-plane are discussed in detail. It is argued that linear boundary conditions that leave the Hamiltonian Hermitian generically break $C$ $P$ and $T$ symmetries as well as Lorentz and conformal symmetry. We show that there is essentially one special case where a single species of fermion has $CPT$ and the full Poincare and conformal symmetry of the boundary. We show that, with doubled fermions, there is a second special case which respects $CPT$ but still violates Lorentz and conformal symmetry. This second special case is essentially the unique boundary condition where the Dirac operator has fermion zero mode edge states. We discuss how the edge states lead to exotic representations of scale, phase and translation symmetries and how imposing a symmetry requirement leads to edge ferromagnetism of the system. We prove that the exotic ferromagnetic representations are indeed carried by the ground states of the system perturbed by a class of interaction Hamiltonians which includes the non-relativistic Coulomb interaction.