By János K. Asbóth, László Oroszlány, András Pályi Pályi
This course-based primer presents novices to the sphere with a concise advent to a few of the center issues within the rising box of topological insulators.
the purpose is to supply a uncomplicated knowing of area states, bulk topological invariants, and of the bulk--boundary correspondence with as easy mathematical instruments as attainable.
the current strategy makes use of noninteracting lattice versions of topological insulators, construction steadily on those to reach from the easiest one-dimensional case (the Su-Schrieffer-Heeger version for polyacetylene) to two-dimensional time-reversal invariant topological insulators (the Bernevig-Hughes-Zhang version for HgTe). In each one case the dialogue of easy toy versions is by means of the formula of the overall arguments concerning topological insulators.
the single prerequisite for the reader is a operating wisdom in quantum mechanics, the proper strong nation physics heritage is supplied as a part of this self-contained textual content, that is complemented via end-of-chapter problems.
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Extra resources for A Short Course on Topological Insulators: Band Structure and Edge States in One and Two Dimensions
46), and a1 , respectively, bN , used to fix the norm of jLi, respectively, jRi. 1 Hybridization of Edge States The two states jLi and jRi hybridize under HO to an exponentially small amount, and this induces a small energy splitting. We can obtain an estimate for the splitting, and the energy eigenstates, to a good approximation using adiabatic elimination of the other eigenstates. N 1/= vN bN ˇ ei ; with some 2 Œ0; 2 /. 54) The energy of the hybridized states thus is exponentially small in the system size.
10) Each internal edge of the lattice is shared between two plaquettes, and therefore occurs twice in the product. However, since we fixed the orientation of the plaquette phases, these two contributions will always be complex conjugates of each other, and cancel each other. Therefore the exponent in the right-hand-side of Eq. 10) simplifies to the exponent appearing in Eq. 11) nD1 mD1 This result is reminiscent of the Stokes theorem connecting the integral of the curl of a vector field on an open surface and the line integral of the vector field along the boundary of the surface.
The motivation is that certain physical parameter spaces in fact have this torus topology, and the corresponding Chern number does have physical significance. kx ; ky C 2 / are equivalent. 38) P As this can be interpreted as a continuum limit of the discrete Chern number, it inherits the properties of the latter: the continuum Chern number is a gauge invariant integer. For future reference, let us lay down the notation to be used for calculating the Chern numbers of electronic energy bands in two-dimensional crystals.