Classification of topological insulators and superconductors in three spatial dimensions (0803.2786v3)

Abstract: We systematically study topological phases of insulators and superconductors (SCs) in 3D. We find that there exist 3D topologically non-trivial insulators or SCs in 5 out of 10 symmetry classes introduced by Altland and Zirnbauer within the context of random matrix theory. One of these is the recently introduced Z_2 topological insulator in the symplectic symmetry class. We show there exist precisely 4 more topological insulators. For these systems, all of which are time-reversal (TR) invariant in 3D, the space of insulating ground states satisfying certain discrete symmetry properties is partitioned into topological sectors that are separated by quantum phase transitions. 3 of the above 5 topologically non-trivial phases can be realized as TR invariant SCs, and in these the different topological sectors are characterized by an integer winding number defined in momentum space. When such 3D topological insulators are terminated by a 2D surface, they support a number (which may be an arbitrary non-vanishing even number for singlet pairing) of Dirac fermion (Majorana fermion when spin rotation symmetry is completely broken) surface modes which remain gapless under arbitrary perturbations that preserve the characteristic discrete symmetries. In particular, these surface modes completely evade Anderson localization. These topological phases can be thought of as 3D analogues of well known paired topological phases in 2D such as the chiral p-wave SC. In the corresponding topologically non-trivial and topologically trivial 3D phases, the wavefunctions exhibit markedly distinct behavior. When an electromagnetic U(1) gauge field and fluctuations of the gap functions are included in the dynamics, the SC phases with non-vanishing winding number possess non-trivial topological ground state degeneracies.

• The paper establishes a comprehensive classification scheme using Altland-Zirnbauer symmetry classes to catalog 3D topological phases.
• It identifies topological invariants, including Z₂ indices and winding numbers, that distinguish trivial from nontrivial phases and protect gapless boundary modes.
• This framework paves the way for experimental advances in quantum computing and the development of novel superconducting materials.

Topological Classification of Insulators and Superconductors in Three Dimensions

The paper entitled "Classification of topological insulators and superconductors in three spatial dimensions" by Schnyder et al. presents a thorough classification scheme for topological phases of non-interacting fermionic systems within certain symmetry classes, based on the framework established by Altland and Zirnbauer. This work thoroughly explores how topology and symmetry interlace to produce novel states of matter, characterizing these phases through homotopy groups, topological invariants, and the gleaned boundary phenomena.

Classification Framework

The classification centers around the ten symmetry classes initially described within the random matrix theory, often referred to as the Altland-Zirnbauer (AZ) class framework. These include the familiar Wigner-Dyson classes and extend to include chiral and Bogoliubov-de Gennes (BdG) classes pertinent to time-reversal, particle-hole, and chiral symmetries. The paper focuses on recognizing 3D topological insulators and superconductors (or superfluids) within five of these classes: AII, AIII, DIII, CI, and CII.

Topological Invariants and Phases

Schnyder et al. describe these systems using topological invariants, like the $Z_2$ invariant for class AII and winding numbers for the chiral classes AIII, DIII, and CI. In class CII, the paper predicts $Z_2$-classified phases. For classes supporting integer winding numbers, these quantities are instrumental in distinguishing between topologically trivial and nontrivial phases of matter. Such invariants serve as haLLMarks for robustness against local perturbations, essentially encoding the topological nature of the ground states.

Boundary Phenomena and Dynamics

A key result of the research is the demonstration that the nontrivial bulk topology ensures the existence of robust gapless modes at the boundaries of these 3D systems. For chiral classes AIII, DIII, and CI, the presence of a sublattice (chiral) symmetry allows the Hamiltonians to be expressed in block off-diagonal form, and their nontrivial topology manifests as stable Dirac or Majorana fermion modes at surfaces. For class CII, a $Z_2$ topological classification protects surface phenomena similar to the behavior of edge states in the quantum spin Hall effect in 2D systems.

Practical and Theoretical Implications

This classification extends the field of topological states beyond the previously explored quantum Hall and $Z_2$ topological insulator phenomena, emphasizing the wide relevance of topology in condensed matter systems. It sets the stage for realizing novel superconducting materials characterized by protected edge states—of particular interest in fields like quantum computing due to their potential for carrying dissipationless currents and manipulating quantum information robustly.

Additionally, the theoretical interrelations with symmetry classes in the context of disorder—through nonlinear sigma models endowed with additional topological terms—foreground new insights into Anderson localization issues. These results suggest connections with experiments probing disorder-induced delocalization at material surfaces, reinforcing the robustness of topological conductors against impurities.

Future Directions

Given the foundational nature of this classification, it lays the groundwork for extensive exploration into interactions beyond non-interacting regimes, where topological quantum matter might unveil further surprises in symmetry-protected phases. Furthermore, the potential for realizing such states in cold atom setups, or through advanced quantum simulations, provides an exciting avenue for experimental realization. These theoretical guidelines bridge the gap towards aspiring applications in robust quantum device technologies, where preserving coherence over disordered and noisy environments is quintessential.

In conclusion, Schnyder et al.'s paper represents a pivotal advancement in our understanding of topological phases in three dimensions, providing a comprehensive framework that intertwines quantum mechanics, topology, and symmetry—a triad that continues to illuminate paths for discovery in condensed matter physics and beyond.