Topological states of matter are a novel family of phases that elude the conventional Landau paradigm of phase transitions. Topological phases are characterized by global topological invariants which are typically reflected in the quantization of physical observables. Moreover, their characteristic bulk-boundary correspondence often gives rise to robust surface modes with exceptional features, such as dissipationless charge transport or non-Abelian statistics. In this way, the study of topological states of matter not only broadens our knowledge of matter but could potentially lead to a whole new range of technologies and applications. In this light, it is of great interest to find novel topological phases and to study their unique properties. In this work, novel manifestations of topological states of matter are studied as they arise when materials are subject to additional symmetries. It is demonstrated how symmetries can profoundly enrich the topology of a system. More specifically, it is shown how symmetries lead to additional nontrivial states in systems which are already topological, drive trivial systems into a topological phase, lead to the quantization of formerly non-quantized observables, and give rise to novel manifestations of topological surface states. In doing so, this work concentrates on weakly interacting systems that can theoretically be described in a single-particle picture. In particular, insulating and semi-metallic topological phases in one, two, and three dimensions are investigated theoretically using single-particle techniques.

Topological states of matter are a novel family of phases that elude the conventional Landau paradigm of phase transitions. Topological phases are characterized by global topological invariants which are typically reflected in the quantization of physical observables. Moreover, their characteristic bulk-boundary correspondence often gives rise to robust surface modes with exceptional features, such as dissipationless charge transport or non-Abelian statistics. In this way, the study of topological states of matter not only broadens our knowledge of matter but could potentially lead to a whole new range of technologies and applications. In this light, it is of great interest to find novel topological phases and to study their unique properties. In this work, novel manifestations of topological states of matter are studied as they arise when materials are subject to additional symmetries. It is demonstrated how symmetries can profoundly enrich the topology of a system. More specifically, it is shown how symmetries lead to additional nontrivial states in systems which are already topological, drive trivial systems into a topological phase, lead to the quantization of formerly non-quantized observables, and give rise to novel manifestations of topological surface states. In doing so, this work concentrates on weakly interacting systems that can theoretically be described in a single-particle picture. In particular, insulating and semi-metallic topological phases in one, two, and three dimensions are investigated theoretically using single-particle techniques.

Robust states exist at the interfaces between topologically trivial and nontrivial phases of matter. These boundary states are expression of the nontrivial bulk properties through a connection dubbed the bulk-boundary correspondence. Whether the bulk is topological or not is determined by the value of a topological invariant. This quantity is defined with respect to symmetries and dimensionality of the system, such that it takes only quantized values. For static topological phases that are realized in ground-states of isolated, time-independent systems, the topological invariant is related to the properties of the Hamiltonian operator. In contrast, Floquet topological phases that are realized in open systems with periodical pumping of energy are topologically characterized with a unitary Floquet operator i.e., the time-evolution operator over the entire period. Topological phases of matter can be distinguished by the dimensionality of robust boundary states with respect to the protecting bulk. This dissertation concerns recently discovered higher-order topological phases where the difference between dimensionalities of bulk and boundary states is larger than one. Using analytical and numerical single-particle techniques, we focus on instances where static higher-order topology can be understood with insights from the mature field of Floquet topology. Namely, even though static systems do not admit a Floquet description, we find examples of higher-order systems to which certain unitary operators can be attributed. The understanding of topological characteristics of these systems is therefore conditioned by the knowledge on topological properties of unitary operators, among which the Floquet operator is well-known. The first half of this thesis concerns toy models of static higher-order topological phases that are topologically characterized in terms of unitary operators. We find that a class of these systems called quadrupole topological insulators exhibit a wider range of topological phases than known previously. In the second half of this dissertation, we study reflection matrices of higher-order topological phases and show that they can exhibit the same topological features as Floquet systems. Our findings suggest a new route to experimental realizations of Floquet systems, the one that avoids noise-induced decoherence inevitable in many other experimental setups.