Topological Insulators

1. Syed Ali Raza Supervisor: Dr. PervezHoodbhoy Topological Insulators

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 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.

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 AharanovBohmEffect 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. AharanovBohm 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 spinorwell 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 diracquantisationcondition 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, 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)