Quasistatic remanence in Dzyaloshinskii-Moriya interaction driven weak ferromagnets and piezomagnets

Topological insulators are electronic materials that have a bulk band gap like an ordinary insulator, but have protected conducting states on their edge or surface. The 2D topological insulator is a quantum spin Hall insulator, which is a close cousin of the integer quantum Hall state. A 3D topological insulator supports novel spin polarized 2D Dirac fermions on its surface. In this Colloquium article we will review the theoretical foundation for these electronic states and describe recent experiments in which their signatures have been observed. We will describe transport experiments on HgCdTe quantum wells that demonstrate the existence of the edge states predicted for the quantum spin Hall insulator. We will then discuss experiments on Bi_{1-x}Sb_x, Bi_2 Se_3, Bi_2 Te_3 and Sb_2 Te_3 that establish these materials as 3D topological insulators and directly probe the topology of their surface states. We will then describe exotic states that can occur at the surface of a 3D topological insulator due to an induced energy gap. A magnetic gap leads to a novel quantum Hall state that gives rise to a topological magnetoelectric effect. A superconducting energy gap leads to a state that supports Majorana fermions, and may provide a new venue for realizing proposals for topological quantum computation. We will close by discussing prospects for observing these exotic states, a well as other potential device applications of topological insulators.

We study the effects of spin orbit interactions on the low energy electronic structure of a single plane of graphene. We find that in an experimentally accessible low temperature regime the symmetry allowed spin orbit potential converts graphene from an ideal two dimensional semimetallic state to a quantum spin Hall insulator. This novel electronic state of matter is gapped in the bulk and supports the quantized transport of spin and charge in gapless edge states that propagate at the sample boundaries. The edge states are non chiral, but they are insensitive to disorder because their directionality is correlated with spin. The spin and charge conductances in these edge states are calculated and the effects of temperature, chemical potential, Rashba coupling, disorder and symmetry breaking fields are discussed.

Since the 1950s Heisenberg and others have attempted to explain the appearance of countable particles in quantum field theory in terms of stable localized field configurations. As an exception Skyrme's model succeeded to describe nuclear particles as localized states, so-called 'skyrmions', within a non-linear field theory. Skyrmions are a characteristic of non-linear continuum models ranging from microscopic to cosmological scales. Skyrmionic states have been found under non-equilibrium conditions, or when stabilised by external fields or the proliferation of topological defects. Examples are Turing patterns in classical liquids, spin textures in quantum Hall magnets, or the blue phases in liquid crystals, respectively. However, it is believed that skyrmions cannot form spontaneous ground states like ferromagnetic or antiferromagnetic order in magnetic materials. Here, we show theoretically that this assumption is wrong and that skyrmion textures may form spontaneously in condensed matter systems with chiral interactions without the assistance of external fields or the proliferation of defects. We show this within a phenomenological continuum model, that is based on a few material-specific parameters that may be determined from experiment. As a new condition not considered before, we allow for softened amplitude variations of the magnetisation - a key property of, for instance, metallic magnets. Our model implies that spontaneous skyrmion lattice ground states may exist quite generally in a large number of materials, notably at surfaces and in thin films as well as in bulk compounds, where a lack of space inversion symmetry leads to chiral interactions.