By János K. Asbóth, László Oroszlány, András Pályi Pályi
This course-based primer offers novices to the sphere with a concise creation to a couple of the middle themes within the rising box of topological insulators.
the purpose is to supply a simple knowing of part states, bulk topological invariants, and of the bulk--boundary correspondence with as basic mathematical instruments as attainable.
the current process makes use of noninteracting lattice versions of topological insulators, development progressively 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 every one case the dialogue of easy toy versions is by way of the formula of the final arguments relating to topological insulators.
the single prerequisite for the reader is a operating wisdom in quantum mechanics, the appropriate stable kingdom physics history is supplied as a part of this self-contained textual content, that's complemented by way of 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
However, their winding numbers differ, and since winding numbers cannot change under adiabatic deformation, we know that they are not adiabatically equivalent. 3 Number of Edge States as a Topological Invariant We have seen in Sect. 2, that the number of edge states at one end of the SSH model was an integer that did not change under a specific type of adiabatic deformation. We now generalize this example. , with length N ! 1, in an energy window from " < E < ", with " in the bulk gap. There can be nonzero-energy edge states in this energy window, and zero-energy edge states as well.
50) If in the thermodynamic limit, the bulk average values, Eq. 51) mD1 with the coefficients am and bm chosen according to Eqs. 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.
D/ D 0 for d d0 but not for d < d 0 . d/i up to a phase factor, has to move simultaneously to zero energy. d0 /i, which occupy sublattice A and B, respectively. Thus, the number NA NB is unchanged. The deformation can also bring a zero energy state j«0 i to energy E > 0 at some d D d 0 . However, it must also create a chiral symmetric partner with energy E < 0 at the same d 0 . This is the time reverse of the process of the previous paragraph: here, both NA and NB must decrease by 1, and, again, NA NB is unchanged.