Massless fermions on a half-space: the curious case of 2+1-dimensions

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.

Automated summary

This article discusses in detail the boundary conditions for a massless Dirac fermion in 2+1 dimensions where the space is a half-plane. It is argued that linear boundary conditions generically break C P, T, Lorentz, and conformal symmetries. However, there are special cases where these symmetries can be maintained. The article also explores the exotic representations and edge ferromagnetism that arise from the edge states.