Quantum Hall effect effect Quantum Hall integer integer Hall bar - PowerPoint PPT Presentation

Quantum Hall effect effect Quantum Hall integer integer Hall bar geometry classical quantum classical Hall effect flux quantum 1 = flux quanta per electron flux per electron

Quantum Hall effect effect Quantum Hall integer integer classical quantum conductivity / resistivity tensor

Quantum Hall effect effect Quantum Hall integer integer Landau levels: symmetric gauge ground state quasiclassical radius (peak): cyclotron orbit integer flux quantum enclosed

Quantum Hall effect effect Quantum Hall integer integer Landau levels & circular potential exactly solvable lifting the degeneracy wave function peak position

Quantum Hall effect effect Quantum Hall integer integer Landau levels & circular potential quasiclassical limit (weak potential) trajectories equipotential lines y random potential landscape x closed and extended trajectories

Quantum Hall effect effect Quantum Hall integer integer islands Potential landscape coastlines as equipotential lines (contour lines) percolating coastlines lakes

Quantum Hall effect effect Quantum Hall integer integer Laughlins argument • uniform magnetic field B through Corbino disk V B • change of Aharonov-Bohm phase through y L • Aharonov-Bohm phase acts on extended x trajectories around the Corbino ring I fixed current single-valued wave function extended state (pure case) shift by one r m integer ''trajectory'' r m-1 conserved flux enclosed by trajectory

Quantum Hall effect effect Quantum Hall integer integer Laughlins argument energy argument V net shift of 1 el from outer to inner edge B y potential energy L x moving electron against electric potential (V) I electromagnetic energy fixed current inductive energy of current loop no energy change for shift by one gauge invariance r m ''trajectory'' r m-1 per filled Landau level

Quantum Hall effect effect Quantum Hall integer integer localized versus extended states extended T Landau levels localized (without spin) clean disorder potential landscape lifts degeneracy degeneracy

Quantum Hall effect effect Quantum Hall fractional fractional fractional QHE integer QHE Störmer, Tsui and Gossard (1982) von Klitzing, Dorda and Pepper (1980)

Properties of metals of metals Properties properties of metal novel phases Fermi surface instability well described by ''free electrons" e.g. Peierls instability Jellium-model (lattice not essential) metal insulator strong renormalization of external Charge Density Wave perturbations: dynamical dielectric function interaction-driven spontaneous symmetry breaking Quantum Hall effect elementary excitations plasmon collective mode continuum electron-hole excitations topological phase 0