A Short Course on Topological Insulators by János K. Asbóth László Oroszlány & András Pályi

Author:János K. Asbóth, László Oroszlány & András Pályi
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

In this chapter, we have provided a formal description of adiabatic pumping in one-dimensional lattices. After identifying the current operator describing particle flow at a cross section of the lattice, we discussed the quasi-adiabatic time evolution of the lower-band states in a two-band model, and combined these results to express the number of pumped particles in the limit of adiabatic pumping. The central result is that the relevant part of the momentum- and time-resolved current carried by the lower-band electrons is the Berry curvature associated to their band.

Problems

5.1. The smooth pump sequence of the Rice-Mele model

For the smoothly modulated Rice-Mele pumping cycle, see (5.2), evaluate the momentum- and time-dependent current density, and the number of particles pumped through an arbitrary unit cell boundary as the function of time; that is, reproduce Fig. 5.2.

5.2. Parallel-transport time parametrization

Specify a parallel-transport time parametrization for the ground state of the two-level Hamiltonian defined by Eqs. (5.1), (5.2), and (a) k = 0 (b) k = π.

5.3. Quasi-adiabatic dynamics with a different boundary condition

In Sect. 5.2.2, we described a stationary state of a quasi-adiabatically driven two-level system, and used the result to express the number of particles pumped during a complete cycle. How does the derivation and the result change, if we describe the dynamics not via the stationary state, but by specifying that the initial state is the instantaneous ground state of the Hamiltonian at t = 0? Is the final result for the number of pumped particles different in this case?

5.4. Adiabatic pumping in multiband models

Generalize the central result of this chapter in the following sense. Consider adiabatic charge pumping in a one-dimensional multi-band system (, where the energies of the first N filled bands () are below the Fermi energy and the energies of the remaining bands () are above the Fermi energy, and the bands do not cross each other. Show that the number of particles adiabatically pumped through an arbitrary cross section of the crystal is the sum of the Chern numbers of the filled bands.

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