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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

- 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.

Other Topological Systems

- Quantum Hall effect
- Fractional Quantum Hall Effect
- Spin Quantum hall effect
- Topological Hall effect in 2D and 3D

Topology

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

Applications

- Majorana fermions
- Quantum Computing
- Spintronics

Thesis Outline (5 Chapters)

- Chapter 1: Adiabatic approximations, Berry phases, relation to AharanovBohm 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.

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 AharanovBohmEffect and Berry Phases
- Berry Phases and Magnetic Monopoles
- How symmetries and conservation laws are effected by Berry's Connection

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 .

Berry Phases

- Pendulum
- Berry 1984
- Proof of adiabatic Theorem
- Phase factors

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

- “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

Berry Connections and Berry Curvature 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

- Berry’s Connection
- Berry’s Curvature
- Berry’s Phase

- Berry connection can never b 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 1984e 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

Aharanov 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 1984Bohm 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.

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

Berry Phases and Magnetic Monopoles 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

- Berry potential from fast variables, Jackiw.
- Source of magnetic field?
- Source of Berry Potential?

- In a polar coordinates parameter space we define a 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 1984spinor 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 spinorwell defined near the south pole and not at the north pole
- Define the spinors in patches

Berry 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 1984potential 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

The vector 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 1984potential 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 diracquantisationcondition from it.

You can also get this result by Holonomy (Wilczek)

How symmetries and conservation laws 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

- 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.

References 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

- Xia, Y.-Q. Nature Phys. 5, 398402 (2009).
- Zhang, Nature Phys. 5, 438442 (2009).
- Fu, L., Kane, C. L. Mele, E. J. Phys. Rev. Lett. 98, 106803 (2007).
- Moore, J. E. Balents, L. Phys. Rev. B 75, 121306 (2007).
- M Z Hasan and C L Kane ,Colloquium: Topological insulators Rev. Mod. Phys.82 30453067 (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)

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