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Bulk-edge correspondence for two-dimensional topological insulators. (English) Zbl 1291.82120
The paper proves the bulk-edge correspondence for 2D topological insulators and independent particles. With this aim, the authors introduce a $$\mathbb Z^2$$ bulk topological invariant equal to the number of pairs of edge states modulo 2. They claim that there is a room for a strict mathematical approach just as for quantum Hall systems, where Laughlin’s argument has gained in precision and detail by the subsequent mathematical discussion. First, the class of bulk, edge, single-particle discrete Schrödinger operators are introduced which describe insulators, topological or otherwise. A general class of 2D single-particle lattice Hamiltonians is considered with and without edge. They are symmetric under fermionic time reversal and potentially describe topological insulators. By this, the periodicity is postulated only in the direction parallel to the edge. The duality is formulated in its most basic version, in a precise though still preliminary form. Then the Hamiltonian is obtained from the Schrödinger operator on the honeycomb lattice modeling graphene in the single-particle approximation and considering two types of boundary conditions (zigzag and armchair). As a result, the authors give a new proof of the absence of edge states for armchair boundary conditions. In order to define the index of bundles on the 2-torus, as well as some auxiliary indices, their sections and transition matrices are considered and they are classified in terms of a $$\mathbb Z^2$$-invariant. As a result, the bulk index is defined and the basic version of duality is stated in a whole. The bulk index is formulated for the specific and more familiar case, where the lattice Hamiltonian is doubly periodic. In this case, the relevant 2-torus is the Brillouin zone. The authors discuss how that index arises from the general one. All the obtained results have a counterpart in the case of quantum Hall (QH) systems. Then, an independent, alternate version of the duality is included which is based on scattering theory and on Levinson’s theorem, as well as on a comparison between the two versions. Then the authors conjecture an analogous alternate version for quantum spin Hall (QHS) systems. Finally, the paper presents some results on the obtained indices, including their comparison with references.

##### MSC:
 82D20 Statistical mechanical studies of solids 81V70 Many-body theory; quantum Hall effect 81U05 $$2$$-body potential quantum scattering theory 82D80 Statistical mechanical studies of nanostructures and nanoparticles 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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