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 ID: 173110
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Slide1
Topological Insulators
Syed Ali RazaSupervisor: Dr. Pervez HoodbhoySlide2
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 outsideSlide3
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.Slide4
Other Topological Systems
Quantum Hall effectFractional Quantum Hall EffectSpin Quantum hall effectTopological Hall effect in 2D and 3DSlide5
Topology
Donut and MugOlympic RingsWave functions are knotted like the rings and can not be broken by continuous changes.Slide6
Applications
Majorana fermionsQuantum ComputingSpintronicsSlide7
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
.Slide8
What I have done so far
The Adiabatic Theorem and Born Oppenheimer approximationBerry phasesBerry Connections and Berry CurvatureSolve 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
H
ow
symmetries
and conservation
laws are
effected
by Berry's ConnectionSlide9
Adiabatic Approximation
Pendulum Born Oppenheimer approximationFast variables and slow variablesThe adiabatic theorem: If the particle was initially in the nth state of Hi then it will be carried to the nth state of Hf
.Slide10
Infinite square wellSlide11
Berry Phases
PendulumBerry 1984Proof of adiabatic TheoremPhase factorsSlide12
Dynamical phase
Geometrical phaseIn parameter spaceExampleObserved in other fields as opticsSlide13
“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 1984Slide14
Berry Connections and Berry Curvature
Berry’s ConnectionBerry’s CurvatureBerry’s PhaseSlide15
Berry connection can never b
e physically observableBerry connection is physical only after integrating around a closed pathBerry 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 exampleSlide16
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.Slide17
Berry 1984 paper
The phase difference is the Berry Phase.Would become clearer in a while.Slide18
Berry Phases and Magnetic Monopoles
Berry potential from fast variables, Jackiw.Source of magnetic field?Source of Berry Potential?Slide19
In a polar coordinates parameter space we define a
spinor for the hamiltonianThe 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
D
efine
the
spinors
in patchesSlide20
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 fluxDirac StringSlide21
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
) Slide22
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.Slide23
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)