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Topological invariants of edge states for periodic two-dimensional models. (English) Zbl 1271.81210

Summary: Transfer matrix methods and intersection theory are used to calculate the bands of edge states for a wide class of periodic two-dimensional tight-binding models including a sublattice and spin degree of freedom. This allows to define topological invariants by considering the associated Bott-Maslov indices which can be easily calculated numerically. For time-reversal symmetric systems in the symplectic universality class this leads to a \({\mathbb Z}_2\)-invariant for the edge states. It is shown that the edge state invariants are related to Chern numbers of the bulk systems and also to (spin) edge currents, in the spirit of the theory of topological insulators.

MSC:

81V70 Many-body theory; quantum Hall effect
19L10 Riemann-Roch theorems, Chern characters
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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References:

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