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).

We study the one-dimensional spin-1/2 Heisenberg chain with competing ferromagnetic nearestneighbor J1 and antiferromagnetic next-nearest-neighbor J2 exchange couplings in the presence of magnetic field. We use both numerical approaches (the density matrix renormalization group method and exact diagonalization) and effective field-theory approach, and obtain the ground-state phase diagram for wide parameter range of the coupling ratio J1/J2. The phase diagram is rich and has a variety of phases, including the vector chiral phase, the nematic phase, and other multipolar phases. In the vector chiral phase, which appears in relatively weak magnetic field, the ground state exhibits long-range order (LRO) of vector chirality which spontaneously breaks a parity symmetry. The nematic phase shows a quasi-LRO of antiferro-nematic spin correlation, and arises as a result of formation of two-magnon bound states in high magnetic fields. Similarly, the higher multipolar phases, such as triatic (p = 3) and quartic (p = 4) phases, are formed through binding of p magnons near the saturation fields, showing quasi-LRO of antiferro-multipolar spin correlations. The multipolar phases cross over to spin density wave phases as the magnetic field is decreased, before encountering a phase transition to the vector chiral phase at a lower field. The implications of our results to quasi-one-dimensional frustrated magnets (e.g., LiCuVO4) are discussed.

An exhaustive classification scheme of topological insulators and superconductors is presented. The key property of topological insulators (superconductors) is the appearance of gapless degrees of freedom at the interface/boundary between a topologically trivial and a topologically non-trivial state. Our approach consists in reducing the problem of classifying topological insulators (superconductors) in d spatial dimensions to the problem of Anderson localization at a (d-1) dimensional boundary of the system. We find that in each spatial dimension there are precisely five distinct classes of topological insulators (superconductors). The different topological sectors within a given topological insulator (superconductor) can be labeled by an integer winding number or a Z_2 quantity. One of the five topological insulators is the 'quantum spin Hall' (or: Z_2 topological) insulator in d=2, and its generalization in d=3 dimensions. For each dimension d, the five topological insulators correspond to a certain subset of five of the ten generic symmetry classes of Hamiltonians introduced more than a decade ago by Altland and Zirnbauer in the context of disordered systems (which generalizes the three well known "Wigner-Dyson'' symmetry classes).Comment: 12 pages, 5 figures; Proceedings of the L.D. Landau Memorial Conference, 200

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