By János K. Asbóth, László Oroszlány, András Pályi Pályi
This course-based primer presents newbies to the sphere with a concise creation to a couple of the middle subject matters within the rising box of topological insulators.
the purpose is to supply a uncomplicated figuring out of aspect states, bulk topological invariants, and of the bulk--boundary correspondence with as basic mathematical instruments as attainable.
the current strategy makes use of noninteracting lattice versions of topological insulators, development steadily 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 types is by means of the formula of the overall arguments relating to topological insulators.
the one prerequisite for the reader is a operating wisdom in quantum mechanics, the suitable sturdy country physics historical past is supplied as a part of this self-contained textual content, that is complemented via end-of-chapter problems.
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Extra info for A Short Course on Topological Insulators: Band Structure and Edge States in One and Two Dimensions
8), Fnm is taken back into Œ ; / by adding a (positive or negative) integer multiple of 2 . 17) Although we proved it here for the special case of a torus, the derivation is easily generalized to all orientable closed surfaces. We focused on the torus, because this construction can be used as a very efficient numerical recipe to discretize and calculate the (continuum) Chern number of a 2-dimensional insulator , to be defined in Sect. 4. R/i, where the R’s are elements of some D-dimensional parameter space P.
4 Chern Number In the discrete case, we defined the Chern number as a sum of Berry fluxes for a square lattice living on a torus (or any other orientable closed surface). Here, we take a continuum parameter space that has the topology of a torus. 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.
2 The Bloch sphere. d/ D d O . This can be identified with a point in R3 nf0g. , on the angles Â and ', as defined in subfigure (a) and in Eq. 63) These eigenstates depend on the direction of the 3-dimensional vector d, but not on its length. d/ jC d i. The choice of the phase factors ˛ and ˇ above corresponds to fixing a gauge. We will now review a few gauge choices. Â; '/ D 0 for all Â; '. 64), we find Â=2 and '=2. There is problem, however, as you can see if you consider a full circle in parameter space: at any fixed value of Â, let ' D 0 !