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Summary

The topological charge of smooth Yang–Mills gauge fields is discussed, describing in particular the SU(2) instanton. This leads to the Adler–Bell–Jackiw anomaly and to θ-vacuum states, which are similar to energy bands in a crystal. We finally discuss the Atiyah–Singer index theorem in the continuum and more explicitly on the lattice.

Summary

The chapter starts with the derivation of effective Hamiltonian for band electrons in magnetic field and continues with discussion of energy levels in magnetic field for massless Dirac fermions. The case of bilayer graphene is also considered. Special attention is paid to a formation of topologically protected zero-energy modes. Using band electrons in magnetic field as an example, a general concept of Berry phase is introduced. Magneto-oscillation effects for two-dimensional Dirac fermions are considered. Quantum Hall effect is discussed, via topological approach by Thouless and coworkers. At the end, we discuss electronic structure in the presence of crossed electric and magnetic fields and the effects of a smooth disorder on Landau levels for massless Dirac electrons.

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19 - Topology of Gauge Fields

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2 - Electron states in a magnetic field

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2 results

19 - Topology of Gauge Fields

Summary

2 - Electron states in a magnetic field

Summary