1. Topological Insulators
Syed Ali Raza
Supervisor: Dr. Pervez Hoodbhoy

2. What are Topological insulators?
Fairly recently discovered electronic phases of matter.
Theoretically predicted in 2005 and 2007 by Zhang, Zahid Hassan and Moore.
Experimentally proven in 2007.
Insulate on the inside but conduct on the outside

3. Conduct only at the surface.
Arrange themselves in spin up or spin down.
Topological insulators are wonderfully robust in the face of disorder. They retain their unique insulating, surface-conducting character even when dosed with impurities and harried by noise.

4. Other Topological Systems
Quantum Hall effect
Fractional Quantum Hall Effect
Spin Quantum hall effect
Topological Hall effect in 2D and 3D

5. Topology Donut and Mug Olympic Rings
Wave functions are knotted like the rings and can not be broken by continuous changes.

6. Applications
Majorana fermions
Quantum Computing
Spintronics

7. Thesis Outline (5 Chapters)
Chapter 1: Adiabatic approximations, Berry phases, relation to Aharanov Bohm Effect, Relation to magnetic monopoles.
Chapter 2: Solve single spin 1/2 particles in a magnetic field and calculate the Berry phase, do it for spin 1 particles (3x3 matrices).
Chapter 3: Understanding Fractional Quantum Hall Effect from the point of view of Berry Phases, the Hamiltonian approach.
Chapter 4: Topology and Condensed Matter Physics
Chapter 5: Understanding Topological Insulators from the point of view of Berry phases and forms.

8. What I have done so far
The Adiabatic Theorem and Born Oppenheimer approximation
Berry phases
Berry Connections and Berry Curvature
Solve single spin 1/2 particles in a magnetic field and calculating the Berry phase
The Aharanov Bohm Effect and Berry Phases
Berry Phases and Magnetic Monopoles
How symmetries and conservation laws are effected by Berry's Connection

9. Adiabatic Approximation
Pendulum
Born Oppenheimer approximation
Fast variables and slow variables
The adiabatic theorem:
If the particle was initially in the nth state of Hi then it will be carried to the nth state of Hf .

10. Infinite square well

11. Berry Phases Pendulum Berry 1984 Proof of adiabatic Theorem
Phase factors

12. Dynamical phase
Geometrical phase
In parameter space
Example
Observed in other fields as optics

13. “A system slowly transported round a circuit will return to it’s original state; this is the content of the adiabatic theorem. Moreover it’s internal clocks will register the passage of time; this can be regarded as the dynamical phase factor. The remarkable and rather mysterious result of this paper is in addition the system records its history in a deeply geometrical way.” – Berry 1984

14. Berry Connections and Berry Curvature
Berry’s Connection
Berry’s Curvature
Berry’s Phase

15. Berry connection can never be physically observable
Berry connection is physical only after integrating around a closed path
Berry phase is gauge invariant up to an integer multiple of 2pi.
Berry curvature is a gauge-invariant local manifestation of the geometric properties
Illustrated by an example

16. Aharanov Bohm Effect Electrons don’t experience any Lorentz force.
No B field outside solenoid.
Acquires a phase factor which depends on B Field.
Difference in energies depending on B field.

17. Berry 1984 paper
The phase difference is the Berry Phase.
Would become clearer in a while.

18. Berry Phases and Magnetic Monopoles
Berry potential from fast variables, Jackiw.
Source of magnetic field?
Source of Berry Potential?

19. In a polar coordinates parameter space we define a spinor for the hamiltonian
The lower component does not approach a unique value as we approach the south pole.
Multiply the whole spinor with a phase.
We now have a spinor well defined near the south pole and not at the north pole
Define the spinors in patches

20. Berry potential is also not defined globally.
A global vector potential is not possible in the presence of a magnetic monopole.
There is a singularity which is equal to the full monopole flux
Dirac String

21. The vector potential does not describe a monopole at the origin, but one where a tiny tube (the dirac string) comes up the negative z axis, smuggling in the entire flux.
As it is spherically symmetric, we can move the dirac string any where on the sphere with a gauge transformation.
Patch up the two different vector potentials at the equator.
The two potentials differ by a single valued gauge transformation and you can recover the dirac quantisation condition from it.
You can also get this result by Holonomy (Wilczek)

22. How symmetries and conservation laws
Abelian and non abelian gauge theories.
When order matters, rotations do not commute then it’s a non abelian gauge theory. e.g SU(3)
Berry connections and curvatures for non abelian cases
How Berry phases effect these laws.
Symmetries hold, modifications have to made for the constants of motion.
Example in jackiw of rotational symmetry and modified angular momentum.

23. References Xia, Y.-Q. Nature Phys. 5, 398402 (2009).
Zhang, Nature Phys. 5, (2009).
Fu, L., Kane, C. L. Mele, E. J. Phys. Rev. Lett. 98, (2007).
Moore, J. E. Balents, L. Phys. Rev. B 75, (2007).
M Z Hasan and C L Kane ,Colloquium: Topological insulators Rev. Mod. Phys (2010).
Griffiths, Introduction to Quantum Mechanics, (2005).
Berry, Quantal phase factors accompanying adiabatic changes. (1984).
Jackiw, Three elaborations on Berry's connection, curvature and phase.
Shankar, Quantum Mechanics.
Shapere, Wilczek, Geometric phases in physics. (1987)