Problem 2 - Cuprate Fermi Surface
(a) Show/ argue that the energy band is described by the function "(k) =
cos ky a) 2 0 (cos(kx + ky )a + cos(kx ky )a
2 (cos kx a +
Following the arguments from class we can write the discrete Schrdinger equation for the
Magnetism
Ferromagnetism has been known since
ancient times, at least to the Greeks and
Chinese
Basic physics: electrons have spin S=1/2.
Sometimes they align
Much more common is antiferromagnetism:
electron spins orient but in a way that adds
up to ze
Atomic magnetism
Splitting of these degeneracies by Coulomb
interactions between electrons is a hard, manyelectron problem!
For isolated atoms, i.e. with spherical symmetry,
some general rules apply. These are called Hunds
rules (Hund was German so I gues
Bose-Einstein
Condensation
Minimum density m
1
3/2
2 3 ~3
r
(3/2)
2
At high temperature, i.e. small , this is
always satised. But below some T it is not.
Denes
2~
k B Tc =
m
2
(3/2)
2/3
What happens for T<T ?
c
~2 2/3
3.31
m
BEC
Below Tc, the chemica
Physics 123B: Homework 3
due January 30, 4pm to Amanda Jones in Kohn Hall 1220 or by email to Prof. Balents
1. IQHE:
Assuming a 2d electron gas with xed 2d electron density n and mass m, and assuming all
electrons are spin polarized (i.e. they are all spi
Physics 123B: Homework 1
due Friday, January 16, 4pm to Amanda Jones in Kohn Hall 1220. Or by email directly to
Prof. Balents.
1. Cuprate Fermi surface: The important energy band of the cuprate high-temperature
superconducting materials can be modeled by
Physics 123B: Homework 4
due February 13, 4pm to Amanda Jones in Kohn Hall 1220 or by email to Prof. Balents
1. Vortices in rotating superuids: As mentioned in class, a rotating bucket of superuid must contain a non-zero density of vortices. You should as
Physics 123B: Homework 5
due February 27, 4pm to Amanda Jones in Kohn Hall 1220 or by email to Prof. Balents
1. Flux quantum: Suppose experimentalists measure the magnetic ux of vortices in a
new superconducting material. They nd this ux is equal to h/4e.
Physics 123B: Homework 5
due February 27, 4pm to Amanda Jones in Kohn Hall 1220 or by email to Prof. Balents
1. Flux quantum: Suppose experimentalists measure the magnetic ux of vortices in a
new superconducting material. They nd this ux is equal to h/4e.
Homework 4 - Solutions
Problem 1 -Vortices in Rotating Superuids
To start, rst notice that the free energy of a system can be written in terms of its superuid density ns
and the wavefunction phase as
Z
c
~ 2 ns
F = d3 r (ns ns )2 +
(r)2
(1)
2
2m
and the s
Physics 123B: Final
Due March 16, 2015, 9:30AM
1. Imagine that electrons have spin S = 1 instead of S = 1/2, so that there are 3 possible spin
states for a single electron. Neglecting the Zeeman interaction, what would be the values of
the Hall conductivi
Physics 123B: Final
Due March 16, 2015, 9:30AM
1. Imagine that electrons have spin S = 1 instead of S = 1/2, so that there are 3 possible spin
states for a single electron. Neglecting the Zeeman interaction, what would be the values of
the Hall conductivi
Homework 2 - Solutions
Carbon Nanotubes
Part 4)
We can write the wavefunction of the graphene sheet within the tight-binding model in the usual way
~ ~
~
(R) = eikR
(1)
Then, to impose periodic boundary condition we want to ensure that our wavefunction sa
Physics 123B: Homework 2
due Friday, January 23, 4pm to Amanda Jones in Kohn Hall 1220
Carbon nanotubes:
Carbon nanotubes are made up of a section of the graphene lattice that has been wrapped
up into a cylinder. You can specify the way the lattice is wou
Curie Susceptibility
Magnetic moment in general is proportional
to spin
= gB S/~
g-factor
(could be a tensor)
g 2 for pure spin
moment
Bohr magneton
e~
B =
= 9.3 10
2me
= 0.671K/T
Magnetic dipole interaction
H=
spin S quantum spin
S2=S(S+1)
H
24
J/T
C