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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.

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The hall states and geometric phase

The Hall States and Geometric Phase

Jake Wisser and Rich Recklau


  • 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
I. The Ordinary and Anomalous Hall Effects

Hall, E. H., 1879, Amer. J. Math. 2, 287

The ordinary hall effect
The Ordinary Hall Effect


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
The Anomalous Hall Effect


“Pressing effect” much greater in ferromagnetic materials

Additional term predicts Hall voltage in the absence of a magnetic field

Anomalous hall data
Anomalous Hall Data

Where ρxx is the longitudinal resistivity and β is 1 or 2

Ii the aharonov bohm effect and berry phase curvature
II. The Aharonov-Bohm Effect and Berry Phase Curvature

Vector potentials
Vector Potentials

Maxwell’s Equations can also be written in terms of vector potentials A and φ

Schr dinger s equation for an electron travelling around a solenoid
Schrödinger’s Equation for an Electron travelling around a Solenoid


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

Vector potentials and interference
Vector Potentials and Interference a Solenoid

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
Experimental Realization a Solenoid

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
Berry Phase Curvature a Solenoid

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
A Classical Analog a Solenoid

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
Anomalous Velocity a Solenoid



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 Symmetry (TRS) a Solenoid

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
The Quantum Hall Effect a Solenoid

  • 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
What changes in the Quantum Hall Effect? a Solenoid

  • 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 effect1
The Quantum Spin Hall Effect a Solenoid

König et, al

What is a topological insulator ti
What is a Topological Insulator (TI)? a Solenoid


Insulating bulk, conducting surface

Breaking trs
Breaking TRS a Solenoid

  • Breaking TRS suppresses one of the channels in the spin Hall state

  • Addition of magnetic moment

  • Cr(Bi1-xSbx)2Te3

Observations a Solenoid

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 a Solenoid