How edge states get ordered and die: the effects of antiferromagnetic ordering in a two-dimensional topological insulator

The research activity of the internship concerned the study of the magnetic ordering effects
in a paradigmatic topological insulator in two dimensions. Topological matter is a thriving eld,
especially because it could yield the missing building blocks for quantum information scale-up.
Topological insulators are a special kind of insulators which are distinct from the usual band
insulators due to topological properties of the Bloch electrons. Such a non-trivial topology
gives rise to many interesting and, in some sense, unusual properties. These systems are insulators,
so they have a gap in their band structure, yet they admit metallic (gapless) modes localized at
the edge of the material: the edge states. The existence of such modes is associated to the precise
quantization of the conductivity in these materials. Moreover, when time-reversal symmetry holds,
the gapless states are also protected from scattering, making them extremely robust with respect
to non-magnetic disorder and weak electronic interactions.
In the presence of strong Coulomb repulsion, electron systems can undergo antiferromagnetic
(AFM) ordering. The corresponding symmetry breaking opens a gap in the band structure of the
material. We investigate the interplay of an emergent AFM ordering, driven by strong electronic
interaction, and topological properties by means of mean- eld theory for an interacting version
of the Bernevig-Hughes-Zhang (BHZ) model. The onset of magnetic ordering, by breaking timereversal
symmetry, is often expected to obliterate the existence of topological states. When this
expectation is not met, exotic states of matter can appear, such as spin Chern insulators,
axionic insulators, etc. We examine the effects of this competition both in bulk geometry and
in a nite stripe geometry endowed with gapless edge states. We characterize the phase diagram
of the model in both geometrical configurations. In particular we show that close to the transition,
the AFM ordering penetrates the stripe from the boundary with a characteristic spatially
modulated pattern of the magnetization. We compare the results for the BHZ model to similar
but topologically trivial models. We conclude that the presence of the gapless edge states in the
topological state prevents the AFM ordering from completely penetrating into the bulk, allowing
for the coexistence of magnetic order and non-trivial topology.