By János K. Asbóth, László Oroszlány, András Pályi Pályi

This course-based primer offers newbies to the sphere with a concise advent to a few of the center subject matters within the rising box of topological insulators.

the purpose is to supply a simple realizing of aspect 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 types of topological insulators, construction 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 each one case the dialogue of straightforward 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 proper strong country physics heritage is equipped as a part of this self-contained textual content, that's complemented by way of end-of-chapter problems.

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**Example text**

54) where the second term is 0 because @k @l D @l @k but jkl D jlk . 55) n0 ¤n where the parameter set R is suppressed for brevity. The term with n0 D n is omitted from the sum, as it is zero, since because of the conservation of the norm, hrn j ni D hn j rni. 58) Act with HO towards the left in Eq. 59) n ¤n This shows that the monopole sources of the Berry curvature, if they exist, are the points of degeneracy. A direct consequence of Eq. 59), is that the sum of the Berry curvatures of O all eigenstates of a Hamiltonian is zero.

59), is that the sum of the Berry curvatures of O all eigenstates of a Hamiltonian is zero. R/ is discrete 38 2 Berry Phase, Chern Number along a closed curve C , then one can add up the Berry phases of all the energy eigenstates. En ˝ ˇ C n0 ˇ rR HO jni ˇ ˛ hnj rR HO ˇn0 ˝ 0ˇ n ˇ rR HO jni ˇ ˛Á hnj rR HO ˇn0 D 0: The last equation holds because a a; b. 5 Example: The Two-Level System So far, most of the discussion on the Berry phase and the related concepts have been kept rather general. In this section, we illustrate these concepts via the simplest nontrivial example, that is, the two-level system.

14) Although the way we derived this above is intuitive, it remains to be shown that this is a consistent definition. From Chap. 2, it is clear that a gauge transformation can only change the bulk electric polarization by an integer. We will show explicitly in Chap. 5 that the change of this polarization in a quasi-adiabatic process correctly reproduces the bulk current. 15) 50 3 Polarization and Berry Phase This operator is useful, because it fully respects the periodic boundary conditions of the ring.