Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
Phys 268r Fall 2013
Problem Set 4  PART II  Solutions
B. Halperin
Solutions to Problem Set 4  Part II
Problem 5.
a)
We have from the denitions:
dxdx (x ) (x ), (x)H (x) = EX,
where H = v px z + cfw_px , f (s) + V (x) + h(x)x . We compute the commutator
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
C H A P T E R
2
The Simple Pendulum
2.1
INTRODUCTION
Our goals for this chapter are modest: wed like to understand the dynamics of a pendulum.
Why a pendulum? In part, because the dynamics of a majority of our multilink robotics
manipulators are simply t
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
Chapter 5
Geodynamics
5.1 Heat flow
Thermally controlled processes within Earth include volcanism, intrusion of igneous rocks, metamorphism, convection within the mantle and outer core, and plate tectonics. The global heat flow
can be measured by measurin
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
Numerical Analysis I
M.R. ODonohoe
References:
S.D. Conte & C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach, Third edition,
1981. McGrawHill.
L.F. Shampine, R.C. Allen, Jr & S. Pruess, Fundamentals of Numerical Computing, 1997. Wiley.
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
Chapter 4
Seismology.
4.1 Historical perspective.
1678 Hooke Hookes Law
(or
)
1760 Mitchell Recognition that ground motion due to earthquakes is related to wave propagation
1821 Navier Equation of motion
1828 Poisson Wave equation
P & Swaves
1885 Rayl
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
Chapter 3
The Magnetic Field of the Earth
Introduction
Studies of the geomagnetic field have a long history, in particular because of its importance for
navigation. The geomagnetic field and its variations over time are our most direct ways to study
the d
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
CHAPTER11
Machines in Membranes
In going on with these Experiments how many pretty Systems
do we build which we soon nd ourselves obligd to destroy!
If there is no other Use discoverd of Electricity this however
is something considerable, that it may help
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
Copyright 2011 University of Cambridge. Not to be quoted or reproduced without permission.
Dynamical Systems
M. R. E. Proctor
DAMTP, University of Cambridge
Michaelmas Term 2011
c
M.R.E.Proctor
2010
(These notes may be copied for the personal use of stude
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
Fractional Quantized Hall Effect
Discovered by Tsui, Stormer and Gossard (1982)
In samples of very high quality, in very strong magnetic
fields, one finds additional plateaus, where
RH1 = e2/h
But is a simple rational fraction, usually with odd
denominat
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
ZeroEnergy Majorana Modes in CondensedMatter Systems
For P268r.
November 2013
Localized zeroenergy Majorana modes
Extra zero energy degrees of freedom that have been
hypothesized to occur at isolated point defects in some
special correlatedelect
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
Phys 268r Fall 2013
Problem Set 5 Solutions  Corrected v2
B. Halperin
Solutions to Problem Set 5
Problem 1.
a)
We have the Hamiltonian H = v p + V (r), and we can write out the momentum term explicitly:
p=
0
px ipy
px + ipy
0
.
We write the wave functio
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
Physics 268r: Topological States of Matter
Fall 2013  Problem Set 1 Solutions
TF: Debanjan Chowdhury
Problem 1:
We are given that Q is a 2N 2N matrix, which we can rewrite as a N N matrix with components
that are themselves 2 2 matrices, i.e.
Qj,k = qjk
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
Physics 268r: Topological States of Matter
Fall 2013  Problem Set 2 Solutions
TF: Debanjan Chowdhury
Problem 1:
(a.) The unpaired electron can be in any spinstate and since the spin commutes with the Hamiltonian, we ignore it hereafter. The electron can
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
Phys 268r Fall 2013
Problem Set 4  Part I  Solutions
B. Halperin
Solutions to Problem Set 4  Part I
Problem 1.
We assume a perfect system, in which the resistance between any two contacts is RQ = h/e2 .
Therefore, more contacts is equivalent to more eq
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
Phys 268r Fall 2013
Problem Set 3 Solutions
B. Halperin
Solutions to Problem Set 3
Problem 1.
We start in a reference frame moving with the lattice at a velocity v = dR/dt, (so that the
problem looks stationary), and mark it as the primed frame. In this f
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
What about the spin degree of
freedom?
Other explanations for the fractional
quantized Hall effect
Fractions other than 1/m or their particlehole
conjugates
Unitary transformation
(Generalized gauge transformation)
trcfw_rj = eleccfw_rj
Chern Simons ma
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
Transport and dynamic properties in
the FermionChernSimons picture.
Results for f=1/2 (with no impurities)
Electron system is compressible.
Fluctuations in the electron density relax very slowly at long
wavelengths. Relaxation rate obeys
q2, for unsc
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
Quantized Hall effects in bilayers and in
systems with spin degeneracy.
J.P. Eisenstein, Exciton Condensation in Bilayer Quantum Hall
Systems (arXiv:1306.0584)
S. M. Girvin and A. H. MacDonald, Multicomponent Quantum
Hall Systems: The Sum of their Parts
Special Topics in Condensed Matter Physics. Topological States of Matter
PHYS 268r

Fall 2013
190 Chapter 5 Life in the Slow Lane: The Low ReynoldsNumber World
PROBLEMS
5.1 Friction versus dissipation
Gilbert says: You say that friction and dissipation are two manifestations of the same
thing. So high viscosity must be a very dissipative situatio