Band Width and Wave localization as a function of Disorder

Links for program, Mathematica Script, etc.:
simluation.c
Mathematica Package
etc

First I'd like to showcase my results using the simulation and band widths all using disorder in increments of 0.02 from 0 to 1 scaling with t, the tunneling energy.

I was a bit unsure at the time if it was physical that the energy of the well could be greater than the value of t. The Bounded data represents the wells that were capped at this value, or in other words generated from a cutoff normal distribution at the ends. I eventually dismissed the idea, and allowed the wells to have energy relative to each one. This is the unbounded data, and seems to fit a quadratic equation given as \(4x^2+x+4\). \(x\) and the band width is on the scale of \(t\). Here's a link to what the distribution tends to look like. 12 Samples were used. These take the longest to solve.

Next I ran a simulation with a sample rate of 50 comparing the width of the wave through the lattice on the same scale of disorder. As you can see, it seems to fit very well with an inverse relationship. Something about the nature of Normal distributions tell me this should have been expected, but I can't remember at the moment.

These next images show both the nature of the simulation, and how the width is changed through out the simulation.

Here a small sample was taken over a short span of time. You can clearly see how at higher disorder, the width converges more rapidly then lower disorder. With no disorder, the width never converges.

And here is nearly the same graph with a much higher sample rate and over a much longer time. You can see by the error bars that even with 500 samples, the wave can behave quite surprisingly as it moves about finding the lowest energy state.

I'll try to run more simulations to get a more accurate Width vs. Disorder plot, but at 50,000 iterations and 500 samples the program ran more than 8 hours so I'll decrease the accuracy by a bit in the code.

If anyone wants to run the code themselves, I've included the current version as source. The inputs can only be changed by editing the file itself, and the output should be redirected to a file. It is formatted so that it can be instantly inserted into excel.

I'd like to thank the people over at Florida State University, from whom I used the Normal Random Number Generators package to do these simulations in C.

I'd like to discuss the actual physical constrains on these types of simulations, and understand the limitations.

Conductivity

Here is what Helen and I presented today.

Here is our paper on Type 1 Superconductors

https://drive.google.com/file/d/0B_n_3zN-4DDWeGo0NGgxRllwMmM/view?usp=sharing

Fresh post! Bound for more discussion and more comments. Three electron three well matrix formulation

*(Don't stare on the picture too long!)

On the left of the picture above you can see the basis I used to create the 20x20 matrix much like the two electron two well system.This also is the paper I derived the matrix ( I have checked it multiple times) with the squares being the overlap integral and the squares with a asterisk are the negative ones. As you can see it's hermitian just as expected from the hamiltonian.

A hard part of this task was to determine when to put a minus sign in front of a transition. I have no idea why we put the minus sign before but I came up with a pattern and it seems like it works. I had a positive contribution I when \( \uparrow,.. \rightarrow ..,\uparrow\) and \( ..,\downarrow \rightarrow \downarrow,.. \) . Negative contribution -I when \( ..,\uparrow \rightarrow \uparrow,.. \) and \( \downarrow,.. \rightarrow ..,\downarrow \) . Also \( \uparrow\downarrow,.. \rightarrow \downarrow,\uparrow \) is negative I and \( \uparrow\downarrow,.. \rightarrow  \uparrow,\downarrow\) is positive I . I know this seems weird but I needed to come up with a pattern that applies in all cases i encountered and that incorporates the pattern used for the 2 electron 2well system .

Since its a huge matrix, I decided to plug in U=4ev and I= 1ev to get numerical values as solutions.(Note:since its a 20x20 matrix, it has 20 eigenvectors and eigenvalues.) Bellow you can see our beautiful hamiltonian H :

0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  1  0  0  0
0  0  0  0  0  0  0  0  0  0  1  0 -1  0  1  0 -1  0  0  0
0  0  0  0  0  0  0  0  0  0 -1  0  0  0 -1  0  0  0  0  0
0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  1  0  0  0  0
0  0  0  0  0  0  0  0  0  0  0 -1  0  1  0 -1  0  1  0  0
0  0  0  0  0  0  0  0  0  0  0  0  0 -1  0  0  0 -1  0  0
0  0  0  0  0  0  0  0  4  0 -1  0 -1  0  0  0  0  0  0  0
0  0  0  0  0  0  0  0  0  4  0 -1  0  1  0  0  0  0  0  0
0  0  0  1 -1  0  0  0 -1  0  4  0  0  0  0  0  0  0  0  0
0  0  0  0  0  1 -1  0  0 -1  0  4  0  0  0  0  0  0  0  0
0  0  1 -1  0  0  0  0 -1  0  0  0  4  0  0  0  0  0  0  0
0  0  0  0  0  0  1 -1  0  1  0  0  0  4  0  0  0  0  0  0
0  0  0  1 -1  0  0  0  0  0  0  0  0  0  4  0  0  0  1  0
0  0  0  0  0  1 -1  0  0  0  0  0  0  0  0  4  0  0  0 -1
0  0  1 -1  0  0  0  0  0  0  0  0  0  0  0  0  4  0  1  0
0  0  0  0  0  0  1 -1  0  0  0  0  0  0  0  0  0  4  0 -1
0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  0  4  0
0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 -1  0 -1  0  4

( to try out your own I and U just use the form above and replace the values accordingly)

Eigenvalue/Eigenvector solutions and comparison with two electron two well system
Just like the 2 electron 2 well case we get two 0 eigenvalues for the eigenvectors:
(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)=\( \downarrow, \downarrow, \downarrow \)
  and
(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0) =\( \uparrow, \uparrow, \uparrow \)

Now lets start from the highest and lowest energy eigenvectors/eigenvalues

The highest energy,  E=5.819 eV I get the state:

(0,0,0,0,0,-0.122,0.244,-0.122,0,0.559,0,-0.509,0,0.509,0,-0.201,0,0.201,0,0)
Eigenvector,\( \mathbf{e}=-0.122 \cdot \uparrow, \downarrow, \downarrow +0.244 \downarrow, \uparrow, \downarrow-0.122\downarrow, \downarrow, \uparrow\)
\( + 0.559 \uparrow \downarrow, \downarrow, .. -0.509 \uparrow \downarrow, .., \downarrow+0.509 \downarrow, \uparrow \downarrow, ... -0.201 ..,\uparrow\downarrow, \downarrow + 0.201 \downarrow,..,\uparrow\downarrow \)

For comparison, in my older post with the derivation of the two electron two well system (also with I=1ev, U=4eV) I got a highest energy eigenvalue E=5eV.
with eigenvector,\(\mathbf{e}= 0.957\uparrow\downarrow,..+0.116 \uparrow, \downarrow -0.116 \downarrow, \uparrow+0.957 ..,\uparrow\downarrow\)

The lowest energy, E=-1.2 eV we get the eigenstate:

(0,0,0,0,0,0.365,-0.730,0.365,0,-0.088,0,-0.228,0,-0.211,0,2.11,0,0)
Eigenvector \( \mathbf{e}= 0.365 \uparrow, \downarrow, \downarrow - 0.730 \downarrow, \uparrow, \downarrow +0.365 \downarrow,\downarrow,\uparrow \)
\(-0.88 \uparrow \downarrow, \downarrow, .. - 0.228 \uparrow \downarrow, .. , \downarrow+0.228 \downarrow, \uparrow \downarrow, .. -0.211 ..,\uparrow\downarrow,\downarrow +0.211 \downarrow,..,\uparrow\downarrow \)

In my old post I also calculated the minimum eigenvalue,eigenvector with an eigenvalue of E=-1eV.
with eigenvector:\(\mathbf{e}=-1/9\uparrow\downarrow,.. +0.5\uparrow, \downarrow -0.5 \downarrow, \uparrow -1/9..,\uparrow\downarrow\)

As you can see that both lowest energy eigenstates do not completely avoid states with coulomb repulsion but still have the lowest energy. However we mentioned in class that for the 2\(e^-) 2well system the energy eigenstate has net spin 0 .  The 3\(e^-) 3well system does not have net spin 0, raising the following question: Is our previous assertion that there is a relationship between states we have obtained for the 2 electron system and anti-ferromagnetism well founded?
*Not sure if important:*
*********************************************************************************
 As a sanity check of my results, I wanted to mention that I did have a state of E=U=4eV for the three electron three well system that has the same energy as a eigenstate of the two electron system.

For 3e 3w system:
E=4eV,
\(\mathbf{e}= -1/2 \uparrow\downarrow,\downarrow,.. + 1/2 \uparrow\downarrow,..,\downarrow\)
\(+1/2 \downarrow, \uparrow\downarrow,.. - 1/2 ..,\uparrow\downarrow,\downarrow\)

For 2e 2w system:
E=4eV  with eigenvector:
=\(\mathbf{e}= \frac{1}{\sqrt{2}}[-(\uparrow\downarrow,..)+(..,\uparrow\downarrow)] \)
=\(.707[-(\uparrow\downarrow,..)+(..,\uparrow\downarrow)]\)
*********************************************************************************
If you are interested in more states other than the ones I already provided, you can read the next part of the post or explore them yourself using the basis and the results of the eigenvalues/eigenvector calculator in the end of the post.

Layout and Comparison of 4 lowest energy eigenstates

E=-1.2 eV

\(\mathbf{e}= 0.365 \uparrow, \downarrow, \downarrow - 0.730 \downarrow, \uparrow, \downarrow +0.365 \downarrow,\downarrow,\uparrow \)
\(-0.088 \uparrow \downarrow, \downarrow, .. - 0.228 \uparrow \downarrow, .. , \downarrow
+0.228 \downarrow, \uparrow \downarrow, .. -0.211 ..,\uparrow\downarrow,\downarrow
+0.211 \downarrow,..,\uparrow\downarrow \)

First note the high amplitude on the state without coulomb repulsion \(-0.730\downarrow, \uparrow, \downarrow\) which is connected via one hop to four coulomb repulsion states \(- 0.228 \uparrow \downarrow, .. , \downarrow\),\(
+0.228 \downarrow, \uparrow \downarrow, ..\),\( -0.211 ..,\uparrow\downarrow,\downarrow\) and
\(+0.211 \downarrow,..,\uparrow\downarrow\). Now note that 0.365 \( \uparrow, \downarrow, \downarrow\) and \(0.365 \downarrow,\downarrow,\uparrow\) have also a considerable amplitude and they do have their corresponding one hop coulomb repulsion pairs \(-0.228 \uparrow \downarrow, .. , \downarrow\),\(-0.211 ..,\uparrow\downarrow,\downarrow\)   and \(0.228 \downarrow,\uparrow\downarrow,..\), \( 0.211\downarrow,..,\uparrow\downarrow \) respectively. The above observations can yield the conclusion that the state \(\downarrow, \uparrow, \downarrow\) is special and has the highest amplitude because of it's ability to tunnel to four states. The overall form of the eigenvector allows tunneling between many states, even ones with coulomb repulsion, since the decrease in kinetic energy via reduction of confinement counteracts the coulomb repulsion. Ultimately note the very small amplitude state \(-0.088 \uparrow \downarrow, \downarrow,..\) that is only connected via one hop to \( \downarrow, \uparrow \downarrow,.. \). This last state will be discussed further under the next lowest state.

E=-1.162eV

\(\mathbf{e}= -0.369 \uparrow,\uparrow,\downarrow + 0.738 \uparrow, \downarrow, \uparrow -0.369 \downarrow, \uparrow, \uparrow\)
\( - 0.214 \uparrow\downarrow,..,\uparrow 0.214 \uparrow,\uparrow\downarrow,.. - 0.214 .., \uparrow\downarrow, \uparrow+ 0.214 \uparrow,..,\uparrow\downarrow \)

Note that this eigenvector is also composed of a high amplitude on the state without coulomb repulsion \( 0.738 \uparrow, \downarrow, \uparrow\) that is connected by one hop to the four states \(- 0.214 \uparrow\downarrow,..,\uparrow, 0.214 \uparrow,\uparrow\downarrow,.. ,- 0.214 .., \uparrow\downarrow, \uparrow\) and \(+ 0.214 \uparrow,..,\uparrow\downarrow\) . The eigenvector also has \(-0.369 \uparrow,\uparrow,\downarrow\) and \(-0.369 \downarrow, \uparrow, \uparrow \) with their one hop double tunneling partners \(0.214 \uparrow,\uparrow\downarrow,..\), \(0.214 \uparrow,..,\uparrow\downarrow\) and  \(- 0.214 \uparrow\downarrow,..,\uparrow\), \(- 0.214 .., \uparrow\downarrow\) respectively. This is strikingly similar to the lowest energy eigenstate but with up and down reversed. These tunneling relationships seem to be the key of what makes these two first states have so much lower energy!(Note:It turns out that including a state that has access to four electron states greately decreases the energy)

Another important note is that the lowest energy state had a small \(-0.088 \uparrow \downarrow, \downarrow,..\) state which seems to be the only striking difference between the two lowest energy states (everything else is the same if you reverse up and down directions). It seems to be the only explanation why the previous state has lower energy than this one. It allows the tunneling to a further state that must decrease the degree of confinement and hence the kinetic energy of the system.

E=-0.494eV

\(\mathbf{e}= 0.664 \uparrow,\uparrow,\downarrow -0.664 \downarrow,\uparrow,\uparrow\)
\(- 0.073\uparrow\downarrow,\uparrow,..-0.164 \uparrow\downarrow,.., \uparrow-0.164 \uparrow,\uparrow\downarrow,..-0.164..,\uparrow\downarrow, \uparrow- 0.164 \uparrow,..,\uparrow\downarrow+0.073 ..,\uparrow,\uparrow\downarrow \)

First note that this energy is higher from the previous one by 0.668 eV and that the two eigenstates discussed earlier have a difference of only 0.038 eV (only around 6% of 0.668eV). The first question to ask is why? So lets see the differences.

In contrast to the states presented previously, it does not include a high amplitude state that can tunnel to four different states via one hop. Also, it looks a lot like the previous eigenvector if one removed the contribution of the high amplitude \( \uparrow, \downarrow, \uparrow \) state and then renormalized accordingly. It can then be inferred that excluding the state that has access to four electron states increases the energy.

 More specifically, the eigenvector includes two high amplitude states, \( 0.664 \uparrow,\uparrow,\downarrow -0.664 \downarrow,\uparrow,\uparrow\) which can tunnel with one step only to two states each. More specifically, \(0.164\uparrow,..,\uparrow\downarrow \),\(0.164 \uparrow,\uparrow\downarrow,..\) and \(-0.164 \uparrow\downarrow,.., \uparrow\),\(-0.164..,\uparrow\downarrow, \uparrow \).

Finally the tiny amplitude contributions \(0.073 ..,\uparrow,\uparrow\downarrow\) and \(- 0.073\uparrow\downarrow,\uparrow,..\) seem to be similar to the small amplitude state in the lowest energy eigenvector . They reduce the energy allowing further tunneling although they include coulomb repulsion!

E=-0.472eV

\( \mathbf{e}=0.669 \uparrow,\downarrow,\downarrow -0.669 \downarrow,\downarrow,\uparrow \)
\(-0.150 \uparrow\downarrow,..,\downarrow -0.150 \downarrow, \uparrow\downarrow,.. -0.166 .., \uparrow\downarrow,\downarrow -0.166 \downarrow,..,\uparrow\downarrow-0.074 .., \downarrow, \uparrow\downarrow \)

The eigenstate above is related to the previous one just like the lowest with the next lowest energy eigenstate. It would be the same as the previous one if you switched the orientation of down and up (also noting that amplitudes are similar) however it differs in the small amplitude states.  This state has only one small amplitude state that includes coulomb repulsion \(-0.074 .., \downarrow, \uparrow\downarrow\), in contrast with the two states \(0.073 ..,\uparrow,\uparrow\downarrow\) and \(- 0.073\uparrow\downarrow,\uparrow,..\) we had before. I think that the 0.22 eV change in energy is solely because the two small amplitude states lower the kinetic energy and avoid coulomb repulsion more efficiently than the single amplitude state.

The two last eigenstates seem to have low energies using the same principle as the first two, tunneling. I think that they do a lousy job compered to the first two eigenstates but still get to have energy lower than \(E_0=0eV\).
Available info and solutions:
This is the basis used 

Important link!

This talk is very interesting, I think, and involves a number of things we have covered to some extent  including localization involving disorder and spontaneous symmetry breaking. It also has a nice introduction to "More is Different", as well as a discussion of non-equilibrium quantum stat mech. This person may be here as a new professor next year. (I think both the intro and then the part that starts about 1/3 of the way through might be most interesting and relevant to us.)
Please comment.
http://ic.ucsc.edu/videoarchive/candidate/rahul-nandkishore.mov

Thoughts on our discussion of the "Theory of Everything" article

I've been thinking more about our discussion today in class, and I think I have a better idea of why I don't like the concept proposed in the paper. The "Theory of Everything" paper suggests that, at a fundamental level, we cannot derive the laws/theories of some higher level processes with the theories of lower ones (as an example I'll stick with, that high temperature superconductivity might be fundamentally unexplainable with Schrodinger's equation and Maxwell's equations).

We talked about not liking it because we want all of the math to line up; that the math and equations for, say, high temperature super-conductivity, should ultimately line up with the equations of Maxwell's equations and Schrodinger's equation. But after thinking about it I'm not sure this the real problem (I at least have) with it. The real problem I think is causation.

The reason we think high-temp superconductivity's (HTS) equations should relate back to Maxwell/Schrodinger equations is that we think HTS should be caused by those more fundamental theories. And as we discussed, if that's not the case, that should just mean Maxwell and Schrodinger aren't the complete story, and we need a more general fundamental theory that will allow us to describe HTS from fundamental principles. What this paper describes, that fundamental principles simply cannot be used to describe higher level ones, seems to me to imply that higher level processes are not caused by lower level ones. If a process is caused by something, we should be able to trace that causation back, or start from fundamentals and show how the fundamentals cause the higher level behavior. My (potentially flawed) idea here is that if one thing causes another, we should, in one way or another (even if it requires something more advanced than our current tool of math), be able to trace that causation through theory.

If that is a correct assumption, that would mean that if HTS cannot be traced back to fundamental theories, that it must not have a cause. It would be a completely spontaneous process that occurs for absolutely no reason. It would mean that if you were to analyze HTS and perform the classic child analysis of continuously asking "why", that if you kept asking why some part of HTS happens the way it does, then you would eventually run into a dead end. Which flies in the face of everything (I think) being a physicist is about. We never let go of that child-like behavior, and we always strive towards some fundamental "why" of the universe. Which is why I'm so unwilling to accept the paper's premise. As far as I know, the one and only truly spontaneous process physicists are willing to accept (and only begrudgingly) is the big bang. From then on, I would think that all things must be caused by something else, and as such that all things can be traced back to more fundamental equations and theories.

I'm really curious about everybody else's opinion on this, and whether or not I might be off base.

Superconductivity.

This will be our last new topic for this quarter.  I haven't thought of much to say yet, but I highlighted a paper on the BCS theory which is linked here.
https://drive.google.com/file/d/0B_GIlXrjJVn4SU5wQnd1MU5BSFU/view?usp=sharing
  Please feel free to comment, question, discuss, etc.

I think that one important thing to be aware of is that it is not so much the phonon aspect that is critical (there are other "mediating bosons"* that can cause pairing and lead to superconductivity). Rather it is the susceptibility of normal state electrons to pairing due to the existence of a Fermi surface, and Fermi-Dirac statistics. This is the rather startling and unexpected thing. Once the electrons pair, one can view the pairs as bosons and the superconductivity as a Bose condensation (into a charged super-fluid state), which is a non-trivial things as well because it involves quantum phase coherence (whatever that means).

Also, here is rough introduction to some of the theory from http://www.superconductors.org/oxtheory.htm: Electrical resistance in metals arises because electrons propagating through the solid are scattered due to deviations from perfect translational symmetry. These are produced either by impurities (giving rise to a temperature independent contribution to the resistance) or the phonons - lattice vibrations - in a solid. In a superconductor below its transition temperature Tc, there is no resistance because these scattering mechanisms are unable to impede the motion of the current carriers. The current is carried in all known classes of superconductor by pairs of electrons known as Cooper pairs. The mechanism by which two negatively charged electrons are bound together is still controversial in "modern" superconducting systems such as the copper oxides or alkali metal fullerides, but well understood in conventional superconductors such as aluminium in terms of the mathematically complex BCS (Bardeen Cooper Schrieffer) theory. The essential point is that below Tc the binding energy of a pair of electrons causes the opening of a gap in the energy spectrum at Ef (the Fermi energy - the highest occupied level in a solid), separating the pair states from the "normal" single electron states. The size of a Cooper pair is given by the coherence length which is typically 1000Å (though it can be as small as 30Å in the copper oxides). The space occupied by one pair contains many other pairs, and there is thus a complex interdependence of the occupancy of the pair states. There is then insufficient thermal energy to scatter the pairs, as reversing the direction of travel of one electron in the pair requires the destruction of the pair and many other pairs due to the nature of the many-electron BCS wavefunction. The pairs thus carry current unimpeded…

The Theory of Everything. More is Different.

Please read and comment.

https://drive.google.com/file/d/0B_GIlXrjJVn4VGliRy1hRUEySjg/view?usp=sharing
https://drive.google.com/file/d/0B_GIlXrjJVn4TmF5SjBsU29iUGc/view?usp=sharing

How do you feel about matrices?

Today while we were calculating some things some people raised some bona fide questions about the matrix approach. I don't want to sweep that under the rug. Let's discuss here how you feel about matrix formulation of quantum physics (for electrons in a lattice). Or anything else that is on your mind.

Physics 155 homework. due Friday.

1a. Write all eigenvectors and their eigenvalues to linear order in t/U (assuming t/U <1). (You can set \(E_o\) to zero if you want. I don't think it it matters, that is, I don't think it influences any eigenvectors and I think it is just added to each eigenvalue (energy).

1b. For each eigenevector, present your understanding of its nature. To what physical situation does it correspond?  Relate that to its energy.

2. Comment on the post "How do you feel about matrices"

3. What else would you like me to ask you? Any suggestions?

4. Extra credit: Is there a possible relationship between states we have obtained for the 2 electron system and anti-ferromagnetism?

Matrix multiplication problem.

The ability to readily visualize a matrix-vector multiplication is of great value in physics. Here is a little problem to test how comfortable you are with matrix-vector multiplication and using that to find eigenvectors and their eigenvalues. Give it a try. (Don't look at anything after the break unless you are stuck.)

Using just guessing and matrix-vector multiplication, find one or more eigenvectors of this matrix:
$$ \begin{matrix} 1 & 1 &1 &1 \\ 1 & 1 & 1 &1 \\  1 & 1 & 1 &1 \\ 1 & 1 & 1 &1 \\ \end{matrix} $$

As you may know, the sum of the eigenvalues is equal to the trace of the matrix. Is there one eigenvector that "exhausts the trace"?

There is new info

in the post showing results from numerical matrix calculations below. (Now called localization length vs disorder.) Please check that out and comment when you have time.

Physics 155 Outline

Here is an outline of our topics and goals for this quarter. Some parts have been changed to be how we should have done it, or how we would like to see it now, in retrospect.

1. Lattice vibrations in spatially periodic systems.  We used a mass-spring type approach to look at the vibrational eigenmodes of a 1-dimensional chain of nucleii. This didn't really fit in with anything else, and it turned out to be confusing, so it is not a point of emphasis.

2. Electron states in a spatially periodic system. For a lattice of atoms, we showed that each atom bound state evolves into a band of N crystal states. This association between atom states and crystal bands is a critical starting point for solid state physics.  With "projection techniques" (appendix A) and approximations, we showed how band state energies depend on an atom state energy, an overlap integral, \(I_{11}\), and a crystal quantum number q (or k).

3. Bands arising from a 3 bound state square well. We looked explicitly at the bands that form for a lattice composed of square wells. We saw that bandwidth depends critically (exponentially) on how far the atomic state wave-function extends outside the well and that the bandwidth of the highest energy band is largest because its atomic state extends furthest.

4. Conductivity of a metal. We discussed how conductivity can be described, for a half-filled band at T=0, as a shifting of occupation boundaries in k-space leading to an imbalance of electron velocities and thus a net current.

5. Specific heat of a metal. We calculated the specific heat of a metal at low T, \(C_v = dE/dT\), using an integral expression for E and taking the derivative of the Fermi function inside the integral and making approximations.

6. Spontaneous magnetization. We created a simplified model for electron-electron interaction and showed that, in the case of a narrow bandwidth, e-e interaction can lead to spontaneous magnetization below a critical temperature, \(T_c\). The driving force for this phase transition and instability is the electron-electron repulsion, which is diminished when electrons spins align. The e-e interaction has to compete with both the band energy and entropy, both of which favor a state with no spin alignment, known as the normal state.

7. Magnetic susceptibility. Using a free energy minimization approach, we showed that magnetic susceptibility is enhanced by e-e interaction even when that interaction is too weak to establish spontaneous magnetization.

8. Matrix approach to band theory. The states of a single band can be obtained by expressing the quantum Hamiltonian as a Hermitian matrix with \( E_o\) on the diagonal and \( I_{11}\) on the off-diagonal. This matrix is written in the "local basis", however, its eigenstates are Bloch states and its eigenvalues are the Bloch state energies that we got before (Appendix A). The matrix approach allows the introduction of disorder and other variation into the crystal.

9. Matrix approach generalized to include electron-electron repulsion. Bloch states extend throughout the crystal and are very useful, but they are basically one-electron states. Sometimes that is not good enough. For example, anti-ferromagnetism and superconductivity cannot be explained in any way using one-electron states. We introduce a model for 2 electrons in a 2 site lattice that includes on-site electron-electron repulsion, U.

10. The nature of superconductivity. ... I get the impression that people really want to learn about superconductivity, even though it might be difficult. Is that true?

11. "More is Different"  "The Theory of Everything" (see attached papers)

Appendix A: Electron eigenstates can be written in "Bloch form", i.e., as a sum involving the particular atomic state with a shifted peak position and systematically varying "phase factor" coefficients. These states are completely itinerant, i.e, the probability density is the same at every site in the crystal. The Bloch form is:
\(\psi_k (x) = \Sigma_n e^{inak} \psi_o (x-na)\).
The projection involves multiplying the expression \( E_k \psi_k (x) = H \psi_k (x)\) on the left by \(\psi_o(x)\) and integrating to get: \(E_k = E_o + 2 I_{11} cos (ak)\), where \( I_{11}\) is a nearest neighbor overlap integral with units of energy.

Two electrons, two wells matrix formulation, Updated with example !!!

https://drive.google.com/file/d/0BwYBTR2Eeem-ck52cTNoU0QtTHM/view?usp=sharing

Eigenvalue and eigenvectors for the 2 electron matrix.

Please post your questions, comments and results related to the 6x6 matrix describing 2 electrons and two sites. Feel free to edit the matrix in here, as well as some eigenvalues and their eigenvectors.

The best initial focus for this is, I believe, to complete the eigenvalue spectrum; that is to find all 6 eigenvalues and to think about their arrangement as a function of energy from lowest to highest. To get an explicit result from which we can begin to understand the essential physics, let's look for a result valid to 1st order in I/U, where I/U is less than one, and also the specific case I = 1 eV and U = 4 eV.

For each eigenvector there is a story. Why does it have the energy it has? How does it acquire that; what is the nature of its eigenvector? All of the eigenvectors have a story. Some of them have really interesting stories. Find them and tell their stories here. 

PS. To give you a starting point to think (and dream and wonder) about this, I believe eigenvectors think mostly about one thing: energy. Some of them are pretty vain and they are focused on their own eigenvalue. (To be fair, some of them think also about relationships, e.g., orthogonality.)

Localization Length vs Disorder

I've done some more studies with time evolution, and placed the results in a publicly accessible google drive directory linked below. In the "disorder" folder is a collection of plots that show the time evolution of the one electron system across a range of "disorder" values (the potential is defined as E_0 + R, where R is a random number from a gaussian distribution with mean of zero and standard deviation of "disorder"). In the moving_I/2 folder are a collection of similar plots, but now with a sloping potential defined as E_0 + R + well*(I/2).

Here is an interesting plot. This shows something we could call the localization length, as a function of disorder. It is calculated with zero sloping potential but with a site disorder as outlined above. The horizontal access is disorder. The quantity on the vertical access is the large-time asymptotic value of the width of the wave-function. Like,  as a function of time, the wave-function gets wider for a while but then it stops getting wider and settles on a particular width, hence we can call that a localization length. (For zero disorder, that would be infinite because it never stops getting wider not matter how long you go int time.)

I am not sure actually how this is defined since a wave-function width less than 1 is a little puzzling. Maybe there is a normalization issue or something? Also, another point is probably to get really accurate values at low disorder one might need to let the wave function evolve for a longer time. I wonder if there is a threshold level of disorder at which this quantity changes from being infinite (no localization) to finite?