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## The Hall States and Geometric Phase

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**The Hall States and Geometric Phase**Jake Wisser and Rich Recklau**Outline**• Ordinary and Anomalous Hall Effects • The Aharonov-Bohm Effect and Berry Phase • Topological Insulators and the Quantum Hall Trio • The Quantum Anomalous Hall Effect • Future Directions**I. The Ordinary and Anomalous Hall Effects**Hall, E. H., 1879, Amer. J. Math. 2, 287**The Ordinary Hall Effect**VH Charged particles moving through a magnetic field experience a force Force causes a build up of charge on the sides of the material, and a potential across it**The Anomalous Hall Effect**VH “Pressing effect” much greater in ferromagnetic materials Additional term predicts Hall voltage in the absence of a magnetic field**Anomalous Hall Data**Where ρxx is the longitudinal resistivity and β is 1 or 2**Vector Potentials**Maxwell’s Equations can also be written in terms of vector potentials A and φ**Schrödinger’s Equation for an Electron travelling around**a Solenoid Where For a solenoid Solution: Where ψ’ solves the Schrodinger’s equation in the absence of a vector potential Key: A wave function in the presence of a vector potential picks up an additional phase relating to the integral around the potential**Vector Potentials and Interference**If no magnetic field, phase difference is equal to the difference in path length If we turn on the magnetic field: There is an additional phase difference!**Experimental Realization**Interference fringes due to biprism Critical condition: Due to magnetic flux tapering in the whisker, we expect to see a tilt in the fringes Useful to measure extremely small magnetic fluxes**Berry Phase Curvature**For electrons in a periodic lattice potential: The vector potential in k-space is: Berry Curvature (Ω) defined as: Phase difference of an electron moving in a closed path in k-space: An electron moving in a potential with non-zero Berry curvature picks up a phase!**A Classical Analog**Non-Zero Berry Curvature Zero Berry Curvature Parallel transport of a vector on a curved surface ending at the starting point results in a phase shift!**Anomalous Velocity**VH E Systems with a non-zero Berry Curvature acquire a velocity component perpendicular to the electric field! How do we get a non-zero Berry Curvature? By breaking time reversal symmetry**Time Reversal Symmetry (TRS)**Time reversal (τ) reverses the arrow of time A system is said to have time reversal symmetry if nothing changes when time is reversed Even quantities with respect to TRS: Odd quantities with respect to TRS:**The Quantum Hall Effect**• Nobel Prize Klaus von Klitzing (1985) • At low T and large B • Hall Voltage vs. Magnetic Field nonlinear • The RH=VH/I is quantized • RH=Rk/n • Rk=h/e2 =25,813 ohms, n=1,2,3,…**What changes in the Quantum Hall Effect?**• Radius r= m*v/qB • Increasing B, decreases r • As collisions increase, Hall resistance increases • Pauli Exclusion Principle • Orbital radii are quantized (by de Broglie wavelengths)**The Quantum Spin Hall Effect**König et, al**What is a Topological Insulator (TI)?**Bi2Se3 Insulating bulk, conducting surface**Breaking TRS**• Breaking TRS suppresses one of the channels in the spin Hall state • Addition of magnetic moment • Cr(Bi1-xSbx)2Te3**Observations**No magnetic field! As resistance in the lateral direction becomes quantized, longitudinal resistance goes to zero Vg0 corresponds to a Fermi level in the gap and a new topological state**References**• http://journals.aps.org/pr/pdf/10.1103/PhysRev.115.485 • http://phy.ntnu.edu.tw/~changmc/Paper/wp.pdf • http://mafija.fmf.uni-lj.si/seminar/files/2010_2011/seminar_aharonov.pdf • https://www.princeton.edu/~npo/Publications/publicatn_08-10/09AnomalousHallEffect_RMP.pdf • http://physics.gu.se/~tfkhj/Durstberger.pdf • http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.5.3 • http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.25.151 • http://www-personal.umich.edu/~sunkai/teaching/Fall_2012/chapter3_part8.pdf • https://www.sciencemag.org/content/318/5851/758 • https://www.sciencemag.org/content/340/6129/167 • http://www.sciencemag.org/content/318/5851/766.abstract • http://www.physics.upenn.edu/~kane/pubs/p69.pdf • http://www.nature.com/nature/journal/v464/n7286/full/nature08916.html • http://www.sciencemag.org/content/340/6129/153