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.
The Hall States and Geometric Phase
Jake Wisser and Rich Recklau
Hall, E. H., 1879, Amer. J. Math. 2, 287
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
“Pressing effect” much greater in ferromagnetic materials
Additional term predicts Hall voltage in the absence of a magnetic field
Where ρxx is the longitudinal resistivity and β is 1 or 2
Maxwell’s Equations can also be written in terms of vector potentials A and φ
For a solenoid
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
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!
Interference fringes due to biprism
Due to magnetic flux tapering in the whisker, we expect to see a tilt in the fringes
Useful to measure extremely small magnetic fluxes
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!
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!
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 (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:
=25,813 ohms, n=1,2,3,…
König et, al
Insulating bulk, conducting surface
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