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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.

We discuss certain aspects of string field theory and its applications in exploring the land- scape of classical string theory vacua. We start by giving a brief overview of various tree-level string field theories, as well as of some relevant mathematical background. As a byproduct of our general discussion of observables, we present a new gauge-invariant quantity for the A∞ formulation of open superstring field theory. Putting particular emphasis on perturba- tive methods, we proceed to review in detail the construction of tree-level effective actions governing the dynamics of a certain subset of degrees of freedom. In light of recent devel- opments, we also discuss efficient methods for evaluating certain vertices of zero-momentum effective actions for open superstring and heterotic string field theories in the presence of a global N = 2 worldsheet superconformal symmetry. We show how to apply this perturbative approach to study dynamics of the D(−1)/D3 system (both with and without a B-field), while also discussing a number of more complicated Dp-brane configurations. At generic points in their moduli spaces, such bound states of Dp-branes clearly cannot be described in terms of simple Dirichlet or Neumann boundary conditions. The rest of this thesis is therefore devoted to developing analytic...

We consider integrable boundary states in the Heisenberg model. We begin by reviewing the algebraic Bethe Ansatz as well as integrable boundary states in spin chains. Then a new class of integrable states that was introduced last year by Caetano and Komatsu is described and expanded. We call these states the crosscap states. In these states each spin is entangled with its antipodal spin. We present a novel proof of the integrability of both a crosscap state that is known in the literature and one that is not previously known. We then use the machinery of the algebraic Bethe Ansatz to derive the overlaps between the crosscap states and off-shell Bethe states in terms of scalar products and other known overlaps. / Vi undersöker integrable gränstillstånd i Heisenbergmodellen. Vi börjar med att gå igenom den algebraiska Betheansatsen och integrabla gränstillstånd i spinnkedjor. Sedan beskrivs och expanderas en ny klass av integrabla tillstånd som introducerades förra året av Caetano och Komatsu. Vi kallar dessa tillstånd crosscap-tillstånd. I dessa tillstånd är varje spinn intrasslat med sin antipodala motsvarighet. Vidare presenterar vi ett nytt bevis av integrerbarheten hos både ett tidigare känt och ett nytt crosscap-tillstånd. Sedan använder vi den algebraiska Betheansatsens maskineri för att härleda överlappen mellan crosscap-tillstånden och off-shell Bethe tillstånd i termer av skalärprodukter och andra kända överlapp.