## PHY 445: Quantum Transport

3 credits | Prerequisites: PHY 203, 304

### Course rationale

Transport phenomena in the nanoscale require the proper applications of Quantum Mechanics. With the advances in technology, new devices are emerging at the nanoscale ushering in the new arena of Nanotechnology. This elective course will focus on the theoretical knowledge required for Nanophysics.

### Course content

Landuer-Buttiker Method: quantum junctions, the Landauer formula, multi-channel scattering and transport, multi-terminal systems; Green Functions: Green functions and the scattering problem, matrix Green functions, advanced and retarded Green functions, Dyson equation, resonant transport; Quantum Tunneling: basics of quantum tunneling, tunneling Hamiltonian method, sequential tunneling; Electron-electron Interaction and Coulomb Blockade: electron-electron interaction in nanosystems, single electron box, single electron transistor, Coulomb blockade in quantum dots, cotunneling; Vibrons and Polarons: electron-vibron interaction in nanosystems, inelastic electron tunneling, local polaron, inelastic tunneling in single-particle approximation, sequential inelastic tunneling; Non-equilibrium Green’s Function: correlation and scattering functions, selfenergy and Green’s function, kinetic equation, calculation of self-energy, relation to Landauer-Buttiker formalism; Schwinger-Keldysh Formalism: non-equilibrium problem, ground-state formalism, closed time path formalism, non-equilibrium diagrammatics, the self-energy; Integer Quantum Hall Effect: classical Hall effect, quantum Hall effect, Landau levels, conductivity in filled Landau levels, robustness of the Hall effect, particles on a lattice; Fractional Quantum Hall Effect: Laughlin states, quasi-holes and quasi particles, anyons, composite fermions, half-filled Landau level; Topology and Berry Phase: adiabatic evolution and the geometry of Hilbert space, Berry phase and the Aharonov-Bohm effect, spin-1/2 Berry phase, Berry curvature of Bloch bands, anomalous velocity, topological quantization; Topological Matter: quantum hall fluid on a torus, hydrodynamic theory, superconductors as topological fluids, topological superconductors, edge states, bulk-edge correspondence, topological insulators, quantum anomalous Hall effect, Z2 topological invariants, topological insulators and interactions.

### Course objectives

- Familiarize with formalism of quantum transport.
- Familiarize with field theory in condensed matter.
- Use Green’s function to calculate different quantities.
- Understand the basics of Schwinger-Keldysh formalism.
- Familiarize with the applications of topology in condensed matter.

### References

- Theory of Quantum Transport at Nanoscale An Introduction (1st edition) by Dmitry A. Ryndyk
- Electronic Transport in Mesoscopic Systems by Supriyo Datta
- Quantum Field Theory of Non-equilibrium States by Jorgen Rammer
- Modern Condensed Matter Physics (1st edition) by Steven M. Grivin, and Kun Yang