Syed Ali Raza
Supervisor: Dr. Pervez Hoodbhoy
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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
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Conduct only at the surface.
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Arrange themselves in spin
up or spin down.
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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.
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Quantum Hall effect
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Fractional Quantum Hall Effect
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Spin Quantum hall effect
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Topological Hall effect in 2D and 3D
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Donut and Mug
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Olympic Rings
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Wave functions are knotted like the rings and
can not be broken by continuous changes.
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Majorana fermions
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Quantum Computing
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Spintronics
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Chapter 1: Adiabatic approximations, Berry phases, relation to
Aharanov Bohm Effect, Relation to magnetic monopoles.
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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).
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Chapter 3: Understanding Fractional Quantum Hall Effect from the
point of view of Berry Phases, the Hamiltonian approach.
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Chapter 4: Topology and Condensed Matter Physics
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Chapter 5: Understanding Topological Insulators from the point of
view of Berry phases and forms.
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The Adiabatic Theorem and Born Oppenheimer approximation
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Berry phases
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Berry Connections and Berry Curvature
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Solve single spin 1/2 particles in a magnetic field and calculating
the Berry phase
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The Aharanov Bohm Effect and Berry Phases
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Berry Phases and Magnetic Monopoles
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How symmetries and conservation laws are effected by Berry's
Connection
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Pendulum
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Born Oppenheimer
approximation
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Fast variables and slow variables
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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 .
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Infinite square well
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Pendulum
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Berry 1984
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Proof of adiabatic
Theorem
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Phase factors
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Dynamical phase
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Geometrical phase
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In parameter space
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Example
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Observed in other fields as
optics
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“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
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Berry’s Connection
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Berry’s Curvature
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Berry’s Phase
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Berry connection can never be physically observable
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Berry connection is physical only after integrating around a
closed path
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Berry phase is gauge invariant up to an integer multiple of
2pi.
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Berry curvature is a gauge-invariant local manifestation of
the geometric properties
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Illustrated by an example
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Electrons don’t
experience any Lorentz
force.
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No B field outside
solenoid.
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Acquires a phase factor
which depends on B Field.
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Difference in energies
depending on B field.
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Berry 1984 paper
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The phase difference
is the Berry Phase.
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Would become
clearer in a while.
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Berry potential from fast variables, Jackiw.
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Source of magnetic field?
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Source of Berry Potential?
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In a polar coordinates parameter
space we define a spinor for the
hamiltonian
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The lower component does not
approach a unique value as we
approach the south pole.
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Multiply the whole spinor with a
phase.
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We now have a spinor well defined
near the south pole and not at the
north pole
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Define the spinors in patches
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Berry potential is also not
defined globally.
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A global vector potential is not
possible in the presence of a
magnetic monopole.
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There is a singularity which is
equal to the full monopole flux
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Dirac String
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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.
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As it is spherically symmetric, we can move
the dirac string any where on the sphere
with a gauge transformation.
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Patch up the two different vector potentials
at the equator.
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The two potentials differ by a single valued
gauge transformation and you can recover
the dirac quantisation condition from it.
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You can also get this result by Holonomy
(Wilczek)
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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.
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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)