Licenciate seminar: Topology Meets Frustration - Exact Solutions for Topological Surface States on Geometrically Frustrated Lattices

One of the main features of topological phases is the presence of robust boundary states that are protected by a
topological invariant. Famous examples of such states are the chiral edge states of a Chern insulator, the helical
edge states of a two-dimensional Z2 insulator, and the Fermi arcs of Weyl semimetals. Despite their omnipresence,
these topological boundary states can typically only be theoretically investigated through numerical studies due to
the lack of analytical solutions for their wave functions. In the rare cases that wave-function solutions are available,
they only exist for simple fine-tuned systems or for semi-infinite systems. Exact solutions are, however, common in
the field of flat bands physics, where they lead to an understanding of the bulk bands rather than the boundary
physics. It is well known that fully-periodic lattices with a frustrated geometry host localized modes that have a
constant energy throughout the Brillouin zone. These localized modes appear due to a mechanism referred to as
destructive interference, which leads to the disappearance of the wave-function amplitude on certain lattice sites.
Making use of this mechanism, it is shown in this licentiate thesis that exact wave-function solutions can also be
found on d-dimensional geometrically frustrated lattices that feature (d – 1)-dimensional boundaries. These exact
solutions localize to the boundaries when the frustrated lattice hosts a topological phase and correspond to the robust,
topological boundary states.
This licentiate thesis revolves around the publication, which describes the method to finding these exact, analytical
solutions for the topological boundary states on geometrically frustrated lattices, which was authored by the author
of this licentiate thesis together with Maximilian Trescher and Emil J. Bergholtz and published in Physical Review B
on August 30, 2017 with the title Anatomy of topological surface states: Exact solutions from destructive interference
on frustrated lattices. An introduction is given on topological phases in condensed matter systems focussing on those
models of which explicit examples are given in the paper: two-dimensional Chern insulators and three-dimensional
Weyl semimetals. Moreover, by making use of the kagome lattice as an example the appearance of localized and
semi-localized modes on geometrically frustrated lattices is elaborated upon. The chapters in this licentiate thesis
thus endeavor to provide the reader with the proper background to comfortably read, understand, place into context
and judge the relevance of the work in the accompanying publication. The licentiate thesis finishes with an outlook
where it is discussed that the method presented in the paper can be generalized to an even larger class of lattices and
can also be applied to find exact solutions for higher-order topological phases such as corner and hinge states.