### Topological Insulators: Dirac Equation in Condensed Matters

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Be the first to write a review About this product. About this product Product Information Topological insulators is insulating in the bulk, but processes metallic states present around its boundary owing to the topological origin of the band structure. The metallic edge or surface states are immune to weak disorder or impurities, and robust against the deformation of the system geometry. This book, the first of its kind on topological insulators, presents a unified description of topological insulators from one to three dimensions based on the modified Dirac equation.

A series of solutions of the bound states near the boundary are derived, and the existing conditions of these solutions are described. Topological invariants and their applications to a variety of systems from one-dimensional polyacetalene, to two-dimensional quantum spin Hall effect and p-wave superconductors, and three-dimensional topological insulators are introduced, helping readers to better understand this fascinating new field.

This book is intended for researchers and graduate students working in the field of topological insulators and related areas.

Additional Product Features Number of Volumes. From the reviews: "The book is devoted to the study of a large family of topological insulators and superconductors based on the solutions of the Dirac equation It is very interesting from a methodical viewpoint for teaching the modern physics of condensed matters.

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## Topological Insulators: Dirac Equation in Condensed Matter by Shun-Qing Shen (Hardback, 2017)

Burn after Writing by Sharon Jones , Paperback 2. Save on Nonfiction Trending price is based on prices over last 90 days. Conceptual Physics,12th Ed. You may also like. By tuning the chemical potential , it is possible to have a precise control of the number of states present, since the density of states varies in a well-defined way with energy. In general, the mass term is the result of breaking a specific symmetry of the system. The size of the band gap can be controlled precisely by controlling the strength of the mass term.

Experimental measurement of the density of states near the Dirac point by standard techniques such as scanning tunnelling microscopy often differ from the theoretical form due to the effects of disorder and interactions. Specific heat, the heat capacity per unit mass, describes the energy required to change the temperature of a sample. Landau quantization refers to the quantization of the cyclotron orbits of charged particles in magnetic fields. As a result, the charged particles can only occupy orbits with discrete energy values, called Landau levels. This key difference plays an important role in the experimental verification of Dirac matter.

For example, the existence of the zeroth Landau level gives rise to a quantum Hall effect where the Hall conductance in quantized at half-integer values. In the context of Fermionic quasiparticles, the Dirac velocity is identical to the Fermi velocity; in bosonic systems, no Fermi velocity exists, so the Dirac velocity is a more general property of such systems. Graphene is a 2-dimensional crystalline allotrope of carbon , where the carbon atoms are arranged on a hexagonal lattice. At these two points, the electronic structure can be effectively described by the Hamiltonian.

There are several distinct ways of introducing a mass term, and the results have different characteristics. A topological insulator is a material that behaves as an insulator in its interior bulk but whose surface contains conducting states.

This property represents a non-trivial, symmetry protected topological order. As a consequence, electrons in topological insulators can only move along the surface of the material. In the bulk of a non-interacting topological insulator, the Fermi level is positioned within the gap between the conduction and valence bands. On the surface, there are special states within the bulk energy gap which can be effectively described by a Dirac Hamiltonian:.

Such Dirac cones emerging on the surface of 3-dimensional crystals were observed in experiment, e.

### Dirac Equation in Condensed Matters

The low-energy properties some semiconducting transition metal dichalcogenide monolayers , can be described by a two-dimensional massive gapped Dirac Hamiltonian with an additional term describing a strong spinâ€”orbit coupling : [12] [13] [14] [15]. Transition metal dichalcogenide monolayers are often discussed in reference to potential applications in valleytronics.

Since all three Pauli matrices are present, there is no further Pauli matrix that could open a gap in the spectrum and Weyl points are therefore topologically protected. In crystals that are symmetric under inversion and time reversal , electronic energy bands are two-fold degenerate. This degeneracy is referred to as Kramers degeneracy.

## [] Topological insulator and the Dirac equation

Therefore, semimetals with linear crossings of two energy bands two-fold degeneracy at the Fermi energy exhibit a four-fold degeneracy at the crossing point. The effective Hamiltonian for these states can be written as. This has exactly the matrix structure of Dirac matter. While historic interest focussed on fermionic quasiparticles that have potential for technological applications, particularly in electronics, the mathematical structure of the Dirac equation is not restricted to the statistics of the particles. This has led to recent development of the concept of bosonic Dirac matter.

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In the case of bosons , there is no Pauli exclusion principle to confine excitations close to the chemical potential Fermi energy for fermions so the entire Brillouin zone must be included. Energy must be added to excite the quasiparticles to the vicinity of the linear crossing point. Several examples of Dirac matter with fermionic quasi-particles occur in systems where there is a hexagonal crystal lattice; so bosonic quasiparticles on an hexagonal lattice are the natural candidates for bosonic Dirac matter. In fact, the underlying symmetry of a crystal structure strongly constrains and protects the emergence of linear band crossings.

Typical bosonic quasiparticles in condensed matter are magnons , phonons , polaritons and plasmons. Anyonic Dirac matter is a hypothetical field which is rather unexplored to date.

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An anyon is a type of quasiparticle that can only occur in two-dimensional systems. Considering bosons and fermions , the interchange of two particles contributes a factor of 1 or -1 to the wave function. In contrast, the operation of exchanging two identical anyons causes a global phase shift. Anyons are generally classified as abelian or non-abelian, according to whether the elementary excitations of the theory transform under an abelian representation of the braid group or a non-abelian one.

The possible construction of anyonic Dirac matter relies on the symmetry protection of crossings of anyonic energy bands. In comparison to bosons and fermions the situation gets more complicated as translations in space do not necessarily commute. Additionally, for given spatial symmetries, the group structure describing the anyon strongly depends on the specific phase of the anyon interchange.

For anyons, an even higher degree of rotation can be necessary, e. From Wikipedia, the free encyclopedia. O; Black-Schaffer, A. M; Balatsky, A. V Advances in Physics. H Solid State Communications. Bibcode : SSCom.

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