We study a link between the ground-state topology and the topology of the lattice via the presence of anomalous states at disclinations -- topological lattice defects that violate a rotation symmetry only locally. We first show the existence of anomalous disclination states, such as Majorana zero-modes or helical electronic states, in second-order topological phases by means of Volterra processes. Using the framework of topological crystals to construct d-dimensional crystalline topological phases with rotation and translation symmetry, we then identify all contributions to (d-2)-dimensional anomalous disclination states from weak and first-order topological phases. We perform this procedure for all Cartan symmetry classes of topological insulators and superconductors in two and three dimensions and determine whether the correspondence between bulk topology, boundary signatures, and disclination anomaly is unique.
We study a link between the ground-state topology and the topology of the lattice via the presence of anomalous states at disclinations -- topological lattice defects that violate a rotation symmetry only locally. We first show the existence of anomalous disclination states, such as Majorana zero-modes or helical electronic states, in second-order topological phases by means of Volterra processes. Using the framework of topological crystals to construct d-dimensional crystalline topological phases with rotation and translation symmetry, we then identify all contributions to (d-2)-dimensional anomalous disclination states from weak and first-order topological phases. We perform this procedure for all Cartan symmetry classes of topological insulators and superconductors in two and three dimensions and determine whether the correspondence between bulk topology, boundary signatures, and disclination anomaly is unique.
Topological phases are quantum phases of matter that appear close to absolute zero temperature. In contrast to commonly known condensed phases of matter, such as a magnet or a superconductor, distinct topological phases may have the same symmetry. Instead, these phases are distinguished by the topology of the electronic wavefunction. Already the single-particle excitations described by band theory in an insulator or a metal or by Bogoliubov-de Gennes theory in a superconductor may have novel anomalous properties associated to their topology. This topological band theory is the subject of this thesis. We aim towards identifying the anomalous boundary and defect states associated to the topology of the band structure in crystalline insulators and superconductors. Furthermore, we determine criteria to easily identify the topological phases from the band structure of a given insulator or superconductor. In Chapter 2 we show that crystalline topological insulators and superconductors may host anomalous gapless or in-gap excitations on corners of a two-dimensional crystal or hinges of a three-dimensional crystal. These topological phases are called "second-order" according to the codimension "2" of their anomalous boundary states. We discuss the precise conditions for their appearance and when they can be linked to the bulk topology. Our discussion includes mirror, twofold-rotation and inversion as crystalline symmetries. In Chapter 3 we discuss the appearance of anomalous gapless or ingap states at defects in a crystalline insulator or superconductor. In particular, we link the existence of anomalous states at disclinations -- lattice defects that violate a rotation symmetry only locally -- to second-order topological crystalline phases. As a side product, we identify the possible contributions of other topological phases to the disclination anomaly. The results of this chapter allow to determine precisely how the disclination anomaly is linked to the bulk topology. In Chapter 4 we present how to construct "symmetry-based indicators" for topological superconductors that admit a description within the Bogoliubov-de Gennes framework. Symmetry-based indicators are necessary criteria for a given Bogoliubov-de Gennes Hamiltonian to realize a topological crystalline phase with anomalous boundary exitations. These criteria are formulated in terms of data from a small set of high-symmetry momenta only. Our approach guarantees to extract the maximal information on the anomalous boundary exitations that can be extracted from a point-wise evaluation of the Bogoliubov-de Gennes Hamiltonian. In the limit of weak superconducting pairing, our criteria can be formulated in terms of data from the normal-state band structure alone.
The imposition of crystalline symmetries is known to lead to a rich variety of insulating and superconducting topological phases. These include higher-order topological phases and obstructed atomic limits with and without filling anomalies. We here comprehensively classify such topological crystalline phases (TCPs) with mirror, twofold rotation, and inversion symmetries in the presence of disorder that preserves the crystalline symmetry on average. We find that the inclusion of disorder leads to a simplification of the classification in comparison to the clean case. We also find that, while clean TCPs evade a general bulk-boundary principle, disordered TCPs admit a complete bulk-boundary correspondence, according to which (bulk) topological phases are topologically equivalent if and only if they have the same anomalous boundary states and filling anomaly. We corroborate the stability of disordered TCPs by way of field-theoretic, numerical, and symmetry-based analyses in various case studies. While the boundary signatures of most disordered TCPs are similar to their clean counterparts, the addition of disorder to certain mirror-symmetric TCPs results in novel higher-order statistical topological phases, in which zero-energy hinge states have critical wave-function statistics, while remaining protected from Anderson localization.
We derive a series of quantitative bulk-boundary correspondences for 3D bosonic and fermionic symmetry-protected topological (SPT) phases under the assumption that the surface is gapped, symmetric, and topologically ordered, i.e., a symmetry-enriched topological (SET) state. We consider those SPT phases that are protected by the mirror symmetry and continuous symmetries that form a group of U(1), SU(2), or SO(3). In particular, the fermionic cases correspond to a crystalline version of 3D topological insulators and topological superconductors in the famous tenfold-way classification, with the time-reversal symmetry replaced by the mirror symmetry and with strong interaction taken into account. For surface SETs, the most general interplay between symmetries and anyon excitations is considered. Based on the previously proposed dimensional reduction and folding approaches, we rederive the classification of bulk SPT phases and define a complete set of bulk topological invariants for every symmetry group under consideration and then derive explicit expressions of the bulk invariants in terms of surface topological properties (such as topological spin, quantum dimension) and symmetry properties (such as mirror fractionalization, fractional charge or spin). These expressions are our quantitative bulk-boundary correspondences. Meanwhile, the bulk topological invariants can be interpreted as anomaly indicators for the surface SETs which carry 't Hooft anomalies of the associated symmetries whenever the bulk is topologically nontrivial. Hence, the quantitative bulk-boundary correspondences provide an easy way to compute the 't Hooft anomalies of the surface SETs. Moreover, our anomaly indicators are complete. Our derivations of the bulk-boundary correspondences and anomaly indicators are explicit and physically transparent. The anomaly indicators obtained in this work can be straightforwardly translated to their time-reversal counterparts that apply to the usual topological insulators and topological superconductors, due to a known correspondence between mirror and time-reversal topological phases.
We show that the gapless boundary signatures---namely, chiral/helical hinge modes or localized zero modes---of three-dimensional higher-order topological insulators and superconductors with inversion symmetry can be gapped without symmetry breaking upon the introduction of non-Abelian surface topological order. In each case, the fractionalization pattern that appears on the surface is ``anomalous'' in the sense that it can be made consistent with symmetry only on the surface of a three-dimensional higher-order insulator/superconductor. Our results show that the interacting manifestation of higher-order topology is the appearance of ``anomalous gapped boundaries'' between distinct topological orders whose quasiparticles are related by inversion, possibly in conjunction with other protecting symmetries such as time-reversal symmetry and charge conservation.
This dissertation discusses extensions of the theory of polarization in crystalline insulators. The notion that electronic wave functions of energy bands in crystalline insulators have Berry phases associated with the positions of electronic charge is at the basis of our understanding of topological band theory. We show, by using the Berry phase formulation and generalizations of it, as well as by exact diagonalization of tight-binding models, that crystalline insulators can generate charge and polarization patterns compatible with the notion of quadrupole and octupole electric moments. In the presence of certain symmetries, these moments are quantized, and their boundary signatures fractionalized. These higher multipole moments then correspond to new symmetry-protected topological phases (SPTs). A salient signature of these SPTs is the existence of zero-dimensional, corner-localized mid-gap energy states. We then study topological crystalline superconductors, in which these zero-dimensional mid-gap states amount to the existence of Majorana bound states (MBS). We thoroughly classify these systems when rotational group symmetries are enforced, and investigate the existence of zero-dimensional MBS trapped at topological defects in these crystals. The finding of these higher multipole moments or generalizations of them to other platforms constitute a new paradigm for the realization of symmetry protected topological phases.
The natural existence of crystalline symmetry in real materials manifests the importance of understanding crystalline symmetry-protected topological phases, especially for interacting systems. In this paper, we systematically construct and classify all the point-group crystalline topological superconductors and insulators in three-dimensional (3D) interacting fermion systems using the novel concept of a topological crystal. The corresponding higher-order topological surface theory can also be systematically studied via higher-order bulk-boundary correspondence. In particular, we discover several intrinsically interacting topological phases that cannot be realized in any free-fermion systems. Moreover, the crystalline equivalence principle for 3D interacting fermionic systems is also verified, with an additional subtle “twist” on the spin of fermions.