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

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Syed ali raza supervisor dr pervez hoodbhoy

Syed Ali Raza

Supervisor: Dr. PervezHoodbhoy

Topological Insulators


What are topological insulators
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
Other Topological Systems

  • Quantum Hall effect

  • Fractional Quantum Hall Effect

  • Spin Quantum hall effect

  • Topological Hall effect in 2D and 3D


Topology
Topology

  • Donut and Mug

  • Olympic Rings

  • Wave functions are knotted like the rings and can not be broken by continuous changes.


Applications
Applications

  • Majorana fermions

  • Quantum Computing

  • Spintronics


Thesis outline 5 chapters
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
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
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
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
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 bohm effect
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
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
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
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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