Dirac materials are fermionic quantum systems in which the low energy physics is captured by the Dirac equation. They are characterized by strong spin-momentum locking and exist in different spatial dimensions. In the first funding period of the SFB 1170, we have investigated the transport properties of one-dimensional (1D) Dirac fermions at the boundary of a 2D topological insulator. In the continu- ation of the project, we want to extend these research activities to higher spatial dimensions. For 2D Dirac fermions, we envision the surface states of 3D topological insulators and, for 3D Dirac fermions, a physical realization is given by the low energy excitations in Weyl or Dirac semimetals. Specifically, we want to consider Dirac systems in a channel geometry which is experimentally accessible and the- oretically rich. The reason for the growing interest in these nanostructures is the interplay of different types of mechanisms – spin-orbit coupling, impurity scattering, scattering off the walls of the channel, electron-electron interactions, and electron-phonon coupling – that influence their transport proper- ties. Interestingly, it is possible to tune the relative magnitude of those mechanisms by a variation of temperature and chemical potential. Hence, a broad range of transport regimes (ballistic, diffusive, hydrodynamic) will be explored.
We aim to analyze the transport properties of Dirac materials by means of a semiclassical Boltz- mann theory. The first step is to derive a proper Boltzmann equation that takes into account the strong spin-momentum locking of Dirac fermions. Technically, we do this by means of a Keldysh-Kadanoff- Baym formalism. We expect to discover new physics on the basis of a refined Boltzmann theory such as a Dirac spin Hall effect. It is interesting to better understand how the presence or the absence of certain symmetries affects the theory. This consideration should help us to establish a connection between our Boltzmann theory valid in the weak coupling regime and AdS/CFT theory that relies on strong Coulomb interactions. The latter theory will be employed by a different team in the SFB 1170. We plan to investigate thermal and electric transport properties and try to identify regimes in which the Wiedemann-Franz law is violated. This violation is typically associated with particular correlations in the system. We want to predict observable quantities – such as electric current, thermal current and their fluctuations – that can be directly measured in transport experiments. Eventually, we envision to obtain a complete understanding of different mechanisms of transport in Dirac materials.
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