Superconductivity: Part 2 by R. D. Parks
By R. D. Parks
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Additional info for Superconductivity: Part 2
2. Again, the spectrum depends crucially on the sign of the gap parameter . The system with the normal band structure ( > 0) remains an ordinary band insulator at subgap energies |E | < , as shown in Fig. 2a. Outside the band gap the conduction- and valence-band states are quantized into discrete edge-state branches evolving into ﬂat bulk Landau levels with increase in q > 0. , propagating in the same direction), the TRS is broken. The results for the inverted band structure with < 0 are qualitatively diﬀerent (see Fig.
5 Mapping the Brillouin zone to the surface of a torus. The cross indicates a zero of the electronic wave function that results in the nontrivial winding of the phase (see also text). • The Hall conductivity is quantized in integer units of the conductance quantum e2 / h. We cannot see this from the CI model for the reasons that will be clear later. Fortunately, the proof rests only on topological properties of the TKNN formula and is free of any model speciﬁcs [45, 48]. 47) we can connect the edges of the Brillouin zone in both kx - and ky - directions, so that it maps to the surface of a torus and, as such, has no boundary (see also Fig.
It allows for a rigorous classiﬁcation of topological phases of an electronic system and is insensitive to continuous deformations of the electronic band structure. For systems with broken TRS, the topological invariant is directly related to an observable, the quantized Hall conductivity. In order to illustrate all these points we shall derive an important formula, ﬁrst proposed by Thouless, Kohmoto, Nightingale, and den Nijs (TKNN) , that expresses the quantum Hall conductivity in terms of the Berry curvature of all occupied electronic bands of the system.