Topological Insulators Syed Ali Raza Supervisor: Dr. Pervez Hoodbhoy

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 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.

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

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 .

Infinite square well

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’s Connection Berry’s Curvature Berry’s Phase

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

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.

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

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

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

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

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)

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.

References 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)