By János K. Asbóth, László Oroszlány, András Pályi Pályi
This course-based primer offers beginners to the sphere with a concise creation to a few of the middle issues within the rising box of topological insulators.
the purpose is to supply a easy figuring out of area states, bulk topological invariants, and of the bulk--boundary correspondence with as easy mathematical instruments as attainable.
the current procedure makes use of noninteracting lattice types of topological insulators, construction progressively on those to reach from the best 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 every one case the dialogue of easy toy versions is by way of the formula of the final arguments concerning topological insulators.
the one prerequisite for the reader is a operating wisdom in quantum mechanics, the correct good nation physics history is supplied as a part of this self-contained textual content, that's 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
Substituting Eqs. 29) into Eq. 11). 3 A Special Case Where the Usual Stokes Theorem Works A shortcut towards a stronger result than Eq. R/i is smooth on the open surface F . 34) 32 2 Berry Phase, Chern Number Summarizing Eqs. R/i is smooth on F , but they might differ with an integer multiple of 2 otherwise. 4 The Case of the Three-Dimensional Parameter Space Let us briefly discuss also the case of a three-dimensional parameter space. This will be particularly useful in the context of two-level systems.
2 Calculating the Berry Curvature and the Berry Phase Consider the two-level system as defined in the previous section. Take a closed curve C in the parameter space R3 nf0g. 72) S where S is any surface whose boundary is the loop C . 59) of the Berry curvature, derived in the last chapter. 61), Eq. 80) We can recognize in this the field of a pointlike monopole source in the origin. Alluding to the analog between the Berry curvature and the magnetic field of electrodynamics (both are derived from a “vector potential”) we can refer to this field, as a “magnetic monopole”.
K/i are analytic functions of k, we have exponential localization of the Wannier states due to properties of the Fourier transform. k/i, the components of the Wannier state jw. 3 Wannier Centers Can Be Identified with the Berry Phase We first assume that we have found a continuous gauge. j/i is hw. 13) The second term in this equation shows that the centers of the Wannier states are equally spaced, at a distance of one unit cell from each other. The first term, which is the Berry phase (divided by 2 ) of the occupied band across the Brillouin zone, cf.
A Short Course on Topological Insulators: Band Structure and Edge States in One and Two Dimensions by János K. Asbóth, László Oroszlány, András Pályi Pályi