We systematically study topological phases of insulators and superconductors (or superfluids) in three spatial dimensions (3D). We find that there exist 3D topologically non-trivial insulators or superconductors in five out of ten symmetry classes introduced in seminal work by Altland and Zirnbauer within the context of random matrix theory, more than a decade ago. One of these is the recently introduced Z2 topological insulator in the symplectic (or spin-orbit) symmetry class. We show there exist precisely four more topological insulators. For these systems, all of which are time-reversal 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. Three of the above five topologically non-trivial phases can be realized as time-reversal invariant superconductors, 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 two-dimensional surface, they support a number (which may be an arbitrary nonvanishing 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 of the Hamiltonian that preserve the characteristic discrete symmetries, including disorder. In particular, these surface modes completely evade Anderson localization from random impurities. These topological phases can be thought of as three-dimensional analogues of well known paired topological phases in two spatial dimensions such as the spinless chiral (px ± ipy)-wave superconductor (or Moore-Read Pfaffian state). In the corresponding topologically non-trivial (analogous to "weak pairing") and topologically trivial (analogous to "strong pairing") 3D phases, the wave functions exhibit markedly distinct behavior. When an electromagnetic U(1) gauge field and fluctuations of the gap functions are included in the dynamics, the superconducting phases with non-vanishing winding number possess non-trivial topological ground state degeneracies.

It has recently been shown that in every spatial dimension there exist precisely five distinct classes of topological insulators or superconductors. Within a given class, the different topological sectors can be distinguished, depending on the case, by a Z or a Z 2 topological invariant. This is an exhaustive classification. Here we construct representatives of topological insulators and superconductors for all five classes and in arbitrary spatial dimension d, in terms of Dirac Hamiltonians. Using these representatives we demonstrate how topological insulators (superconductors) in different dimensions and different classes can be related via "dimensional reduction" by compactifying one or more spatial dimensions (in "Kaluza-Klein"-like fashion). For Ztopological insulators (superconductors) this proceeds by descending by one dimension at a time into a different class. The Z 2 -topological insulators (superconductors), on the other hand, are shown to be lower-dimensional descendants of parent Ztopological insulators in the same class, from which they inherit their topological properties. The 8-fold periodicity in dimension d that exists for topological insulators (superconductors) with Hamiltonians satisfying at least one reality condition (arising from time-reversal or charge-conjugation/particle-hole symmetries) is a reflection of the 8-fold periodicity of the spinor representations of the orthogonal groups SO(N ) (a form of Bott periodicity). Furthermore, we derive for general spatial dimensions a relation between the topological invariant that characterizes topological insulators and superconductors with chiral symmetry (i.e., the winding number) and the Chern-Simons invariant. For lower dimensional cases, this formula relates the winding number to the electric polarization (d = 1 spatial dimensions), or to the magnetoelectric polarizability (d = 3 spatial dimensions). Finally, we also discuss topological field theories describing the space time theory of linear responses in topological insulators (superconductors), and study how the presence of inversion symmetry modifies the classification of topological insulators (superconductors).

Topological materials have become the focus of intense research in recent years, since they exhibit fundamentally new physical phenomena with potential applications for novel devices and quantum information technology. One of the hallmarks of topological materials is the existence of protected gapless surface states, which arise due to a nontrivial topology of the bulk wave functions. This review provides a pedagogical introduction into the field of topological quantum matter with an emphasis on classification schemes. We consider both fully gapped and gapless topological materials and their classification in terms of nonspatial symmetries, such as time-reversal, as well as spatial symmetries, such as reflection. Furthermore, we survey the classification of gapless modes localized on topological defects. The classification of these systems is discussed by use of homotopy groups, Clifford algebras, K-theory, and non-linear sigma models describing the Anderson (de-)localization at the surface or inside a defect of the material. Theoretical model systems and their topological invariants are reviewed together with recent experimental results in order to provide a unified and comprehensive perspective of the field. While the bulk of this article is concerned with the topological properties of noninteracting or mean-field Hamiltonians, we also provide a brief overview of recent results and open questions concerning the topological classifications of interacting systems.

We study the quasi-one-dimensional limit of the spin-1/2 quantum Heisenberg antiferromagnet on the kagome lattice. The lattice is divided into antiferromagnetic spin chains ͑exchange J͒ that are weakly coupled via intermediate "dangling" spins ͑exchange JЈ͒. Using one-dimensional bosonization, renormalization-group methods, and current algebra techniques, the ground state is determined in the limit JЈ Ӷ J. We find that the dangling spins and chain spins form a spiral with O͑1͒ and O͑JЈ / J͒ static moments, respectively, atop of which the chain spins exhibit a smaller O͓͑JЈ / J͒ 2 ͔ antiferromagnetically ordered component along the axis perpendicular to the spiral plane.with J 1 Ͻ 0 and J 2 Ͼ 0, in accordance with Eq. ͑30a͒. The estimated magnitudes of further-neighbor interactions arewhere J 3 is ferromagnetic and J 4 is antiferromagnetic. As anticipated from the RG treatment in Sec. III B, we observe SPATIALLY ANISOTROPIC HEISENBERG KAGOME… PHYSICAL REVIEW B 78, 174420 ͑2008͒ 174420-7