While the importance of spin-orbit coupling in determining magnetic properties has long been appreciated, it was thought until recently that the effects of spin-orbit coupling in non-magnetic materials are relatively minor. A dramatic counterexample is provided by the recently discovered "topological insulators": these are materials in which spin-orbit coupling leads to protected metallic surface or edge states as a consequence of the topology of the bulk electronic wavefunctions. Bulk topological insulators such as Bi_{2}Se_{3} and Bi_{2}Te_{3} have surface states that bear a close resemblance to the Dirac fermion states in graphene, but with the property of spin-helicity which makes them immune to scattering by disorder as well as some other unique features. Without exaggerating, these materials realize new topological electronic phases. The experimental discoveries of two-dimensional topological insulators (the "quantum spin Hall effect") in 2007 and of bulk topological insulators in 2008 have triggered an explosion of work in this field.

Electronic structure methods have already had some notable successes in this field, e.g., in the theoretical prediction that Bi_{2}Se_{3} and Bi_{2}Te_{3} would be topological insulators. We believe that these successes are just the beginning and that there is a considerable opportunity for first-principles methods to contribute to the discovery of not just more topological insulators but also "Chern insulators", which would show an intrinsic quantum Hall effect, and topological antiferromagnets or other correlated systems. There has also been a flow of ideas from topological insulators to more general materials: for example, it is now understood how to compute the orbital contribution to the magnetoelectric polarizability of a general material, building on previous work on the quantized magnetoelectric polarizability in topological insulators. First-principles approaches will certainly play an instrumental role in understanding the effects of nanostructuring, reduced dimensions and disorder in topological insulators.

Bulk topological insulators are electronic materials that have a bulk band gap, but also exhibit conducting surface states immune to scattering. Existence of topological insulating electronic phases in systems with spin-orbit interaction was theoretically predicted only a few years ago [1,2]. The first experimental confirmation of topological phases in two dimensions has been reported for the HgCdTe quantum well structures using transport measurements [3], again guided by a precise theoretical prediction [4]. One year later, photoemission spectroscopy revealed the three-dimensional topological nature of Bi_{1-x}Sb_{x} alloys [5]. Then, a “second generation” of bulk topological insulators, Bi_{2}Se_{3} and Bi_{2}Te_{3}, were found among well known materials of technological importance (see Refs. 6-8 and several other works in 2009). The Bi_{2}X_{3} materials show topologically protected behavior in ordinary crystals at room temperature and zero magnetic field. The topologically protected surface states are characterized by “Dirac cone” dispersion as in graphene, but show no spin-degeneracy. As a result, the surface state charge carriers are immune to scattering and their spin shows one-to-one correspondence with their momentum (helicity). Another way to view bulk topological insulators is an extension of the 2D quantum spin Hall insulators to three dimensions. The range of possible technological applications of this novel state of matter is hard to overestimate: thermoelectrics, “next generation electronics” - spintronics and quantum information processing - to name just a few. It is also anticipated that topological insulators are actually fairly common among the heavy element materials. Many new materials are expected to join this rapidly expanding field of physics and, hopefully, future technology (see Refs. 9-10 for overview).

Historically, most current understanding in the field of topological insulators is provided by phenomenological approaches based on model Hamiltonians. However, there is a large gap in knowledge which cannot be covered using this methodology. For instance, empirical theories are practically useless for guiding the search of novel topological insulating compounds. This gap has to be covered using first-principles approaches.

In fact, first-principles electronic structure calculations have already played an important role in the discovery of Bi_{2}X_{3} topological insulators [6,7]. More recently, topological electronic phases of closely related thallium-based ternary chalcogenides have been predicted from first principles [11,12] and then confirmed experimentally [13]. Even more striking is the theoretical prediction of a number of multifunctional topological insulators belonging to the family of Heusler compounds [14,15]. The binary bismuth chalcogenides Bi_{2}X_{3} are layered materials; nanostructures such as thin films and nanoribbons have been produced experimentally [16]. Significant theoretical efforts are now devoted towards understanding the effects of reduced dimensionality in bulk topological insulators [17-19] and possible technological applications of such nanostructures [20]. Topological insulators doped by magnetic ions constitute another interesting class of materials synthesized experimentally [21]. A recent first-principles study suggests that such materials may realize the quantized anomalous Hall effect [22].