Apr 29 2013

Moving out

Category: Miscellaneous — Andrés G. Saravia @ 4:28 pm

This notebook has been very useful to me in the last months, however it proved to be too… small. I wanted a better way to organize my research and build collaborations. After searching the web for some a better option and not finding it, I decided to make my own tools. Thus Research Engine was born.

I have there not only a notebook like this one (ok, a simpler version of this one) but many notebook, a wiki, a forum, a datasets’ organizer, a collaborative writings tool and some more things. It’s also a web application made entirely by me so I’m frequently improving it to meet my needs.

You can see my work here. If you want to follow my work, do it there. I promise it will be much better. If you want to collaborate with me contact me and we can talk.

Oct 04 2012

Monthly report: September 2012

Category: Monthly report — Andrés G. Saravia @ 10:36 am

We made some interesting things this month:

  • I projected three states of the O-Cu-O cluster model into the electron occupation basis. It’s interesting to notice that the greatest difference between the base state and the first infrared-excited state is in the middle coupling regime.
  • I also plotted the eigenvalues of the cluster without phonons as a function of the on-site coupling (U) to reassure ourselves that there were no crossings nor any funny behavior.
  • I made a brief post to gather all the information and start writing an article! :)
  • Along with the writing of this article I’m taking a course about Writing in the Sciences which should be particularly useful right now. If you don’t know about Coursera I strongly suggest you take a look. It’s awesome!
  • This month I finished an Udacity course on Software Debugging. Here is my certificate. If you don’t know about Udacity, take a look! It’s also awesome!
  • Thanks to the online courses I’ve taken I have now a basic ability to design and implement web application so I’m starting Research Engine as an effort to aggregate in one simple place all the resources I use on my research. This app is intended to be free (as in freedom) and available to every one. If you are interested have a look at the README file in its github repository. It’s in (super) alpha state but I will continue steadily to develop it.

Sep 26 2012

Recapitulation of results on the O-Cu-O cluster

Category: Superconductivity — Andrés G. Saravia @ 2:58 pm

I will make here an index of the ideas we have found in the last year about our quantum model for the O-Cu-O cluster in the YBCO superconductor.

Methodology development: First, we stated the system’s hamiltonian and I implemented a matrix-building and diagonalization algorithm to find the eigenvalues. Using a classical model for the phonon vibrations we could propose a dependence of the phonon frequencies with the atom’s mass (I found a small inconsistency but we think it’s not a great problem). Trying to be as consistent as possible with the experimental values we calculated the change in the coupling constants due to the isotopic shift of Oxygen. We could write the wavefunction in terms of “infrared” and “Raman” coordinates by approximating the polaron behavior as quantum harmonic oscillators and summing over electronic degrees of freedom.

Reproducing previous work: First, we wanted to reproduce the results from PRL 68, 3236; the parameters used are here. We found the same eigenvalues for the lowest-energy states as function of the electron-infrared phonons coupling  \lambda_{ir} (this reproduces Fig. 2) however, looking closely at the data we identified each state a little differently. We also found the same dependence of the mean infrared phonons in the base state here (this reproduces Table 1). We also reproduced the peak splitting of the wavefunction in terms of phonon coordinates when  \lambda_{ir} increases. We also took the parameters used in J. Sup. Nov. Mag. 20, 603 and reproduced Fig. 1 which are the eigenvalues and the isotopic shift in Fig. 5 as functions of \lambda_{ir}.

New results: We found that the Raman coupling constant played a small role in the eigenvalues and the isotopic shift, so we set it to zero for the next calculations since that makes easier the interpretation. With this we could clearly identify the character of the lowest eigenstates according to the number of phonons it has when \lambda_{ir}=0. Interestingly we found a state well above the ground state which seemed electronic in origin. To investigate further I made a purely electronic hamiltonian and found this state as the first excited state. I could even find a simple analytic expression for its eigenvalue: \epsilon - \sqrt{\epsilon^2 + 2t^2}. It’s worth noticing that it’s independent of the on-site repulsion U, this is reinforced by noticing that it hasn’t any component in states with electrons in double occupancy. I tried to understand the dropping in energy of this electronic state as \lambda_{ir} increases thinking of two hopping energies stemming from the two bond distances that we think are present in this regime, however I’m still unsure of how to relate those hopping energies with the change in distance so I’ve found nothing useful so far. Perhaps I should work more on this.

Later we found that this electronic state has an interesting negative isotopic shift peaking in the middle coupling regime (there seems to be a quirk in the calculation at  which should be examined more closely). The wavefunction projected into phonon coordinates splits into two peaks similar to the behavior of the base state, it also has similar mean infrared phonons (the difference peaks in the intermediate regime). Later we summed over the phononic degrees of freedom and got the projection into electron occupation states for the base, the first infrared and the electronic states. We can see that they are somewhat similar and, again, the main difference is in the intermediate coupling regime. Last of all, I looked into the different eigenstate of the purely electronic hamiltonian and took a look at their energies as a function of the on-site repulsion just to be sure that the base state remains the same near the on-site repulsion value (U = 7eV) we are using; this seems to be the case.

We started to investigate the same cluster model with slower electrons (i. e. a smaller hopping energy) but didn’t get very far. Perhaps it will be interesting to look again at this regime later. (link1, link2, link3, link4, link5)

Sep 07 2012

Effect of on-site repulsion in the O-Cu-O cluster with no phonons

Category: Polarons — Andrés G. Saravia @ 10:22 am

Some time ago we found that the first excited state in our O-Cu-O cluster with no phonons had a simple representation and didn’t depend on the parameter U (representing the on-site Coulomb repulsion between two electrons on the same site). So, it’s not evident that the base state will always depend on U, perhaps at high energies there is a level crossing and the U-independet first excited state becomes the base state. In this post I explore such a posibility.

My tool of choice for this is, again, Sage (if you don’t know it, give it a try! it’s awesome). First, I define some variables, write the hamiltonian and calculate the eigenvalues

var('epsilon,U,t')

H = matrix([
[U + 2*epsilon,t,0,t,0,0,0,0,0],
[t,0,t,0,t,0,0,0,0],
[0,t,2*epsilon,0,0,t,0,0,0],
[t,0,0,0,t,0,t,0,0],
[0,t,0,t,U - 2*epsilon,t,0,t,0],
[0,0,t,0,t,0,0,0,t],
[0,0,0,t,0,0,2*epsilon,t,0],
[0,0,0,0,t,0,t,0,t],
[0,0,0,0,0,t,0,t,U+2*epsilon]])

eigval = H.eigenvalues()

Using the parameters we’ve been working on, we can have a look at the numerical values of the eigenvalues

for e in eigval:
    print real(n(e.subs(epsilon=0.5, t=0.5, U=7)))

-0.366025403784439 
1.36602540378444 
-0.536881634602406 
1.31240293602837 
6.15791772613331 
8.06656097244073 
-0.0620192023179804 
8.06201920231798 
0.000000000000000

As we can see, the base state is eigval[2] and the first excited state is eigval[0]. Furthermore, the energy of the first excited state, in cm-1 is

real((eigval[0].subs(epsilon = 0.5, t = 0.5, U = 7)-eigval[2].subs(epsilon = 0.5, t = 0.5, U = 7))*8056)

1376.41779546955

Which is exactly what I expected. To make a plot of the base state and the first excited state I use this simple loop

p_list_2 = []              # The base state
p_list_0 = []              # The first excited state

for u in [0.0, 0.1.., 10]:
    x2 = real(n(eigval[2].subs(epsilon = 0.5, t = 0.5, U = u)))
    x0 = real(n(eigval[0].subs(epsilon = 0.5, t = 0.5, U = u)))
    p_list_2.append([u, x2])
    p_list_0.append([u, x0])

And now we can make a plot

plot2 = list_plot(p_list_2,gridlines=True,legend_label='base state', rgbcolor='black')
plot0 = list_plot(p_list_0,gridlines=True,legend_label='first excited state', rgbcolor='red')
p = plot2 + plot0
p.axes_labels(['U (eV)','eigenvalue (eV)'])
p.xmin(0)
p.xmax(10)
p.ymin(-2)
p.ymax(0)
p.set_legend_options(loc='center right')
show(p)

So, it seems that there is no level-crossing between these two states but they seem to overlap when U \rightarrow \infty. In fact, altough the expression for the base state is complicated we can calculate the limit when U \rightarrow \infty. Altough it takes a long time, Sage can perform this calculation simbolically

eigval[2].limit(U = oo).show()

\epsilon - \sqrt{\epsilon^2 + 2t^2}

So, indeed, they converge.

Just to be sure there is not any other state crossing eigval[2] I also made a plot of all 9 energy levels:

I think it’s interesting that there are degeneracies when U = 0 and U \rightarrow \infty and also that there are three states which seem to be completely independent of U. As I mentioned earlier, some eigenvalues have a simple analytic representation, in fact, 5 of them do

eigval[0].show()  \rightarrow \epsilon - \sqrt{\epsilon^{2} + 2 \, t^{2}}

eigval[1].show()  \rightarrow \epsilon + \sqrt{\epsilon^{2} + 2 \, t^{2}}

eigval[6].show()  \rightarrow \frac{1}{2} \, U + \epsilon - \frac{1}{2} \, \sqrt{U^{2} + 4 \, U\epsilon + 4 \, \epsilon^{2} + 8 \, t^{2}}

eigval[7].show()  \rightarrow \frac{1}{2} \, U + \epsilon + \frac{1}{2} \, \sqrt{U^{2} + 4 \, U\epsilon + 4 \, \epsilon^{2} + 8 \, t^{2}}

eigval[8].show()  \rightarrow 0

The other 4 states have a much more complicated expression.

In conclusion, we don’t have to worry about an energy crossing and the base state will always depend on U, altough much less at large values.

Sep 05 2012

More projections into electronic occupation basis

Category: Uncategorized — Andrés G. Saravia @ 3:37 pm

In my previous post I suggested that it would be a good idea to look also at the projection into the electron occupation basis of the first infrared-excited state. Here it is!

The data files are here:

It is, perhaps, not surprising that the infrared state converges to the base state at large \lambda_{ir} but I'm not sure there was a way to anticipate that both will behave identically at \lambda_{ir}=0 (the proyections are exactly the same). Again we see that most interesting range is the intermediate regime where the difference between both states is largest.

Sep 03 2012

Electronic state projected into the electronic occupation basis.

Category: Uncategorized — Andrés G. Saravia @ 5:04 pm

Previously we looked at the projection of the electronic state without phonons into the electronic occupation basis, we are now interested in seeing how this probabilities change with the \lambda_{ir} coupling. To calculate those projections we need to sum over the phononic degrees of freedom, this is easily done defining the following fucniton inside the MARKKU package:

(defun proyect-in-el-occ-basis (system eigenvector e1 e2)
  (let ((result 0))
    (dotimes (ir (1+ (ir-phonons system)))
      (dotimes (ram (1+ (raman-phonons system)))
	(incf result
	      (funcall (lambda (x) (* x x)) 
		       (matrix-ref (eigenvectors system)
				   (markku.quantum::state-label system 
								(list :e1 e1 :e2 e2 :ir ir :Raman ram))
				   eigenvector)))))
    result))

Defining the appropiate systems and finding which eigenvector number corresponds to the electronic state we can easily find all the relevant projections. Here are the files I obtained for the base sate and the electronic state. I made the following plot (with this gnuplot script) comparing them.

Perhaps it would also be interesting to look at the equivalent projections of the first infrared state…

Sep 03 2012

Monthly report: August 2012

Category: Monthly report — Andrés G. Saravia @ 10:40 am

This month I made a couple of interesting things, first, regarding the polaronic model we've been working on:

  • We found here a way to express the wavefunction in terms of infrared and Raman coordinates (defined as if they were harmonic oscillators) and I implemented it here on Markku.
  • As a test we were able to reproduce here Fig. 2 from PRL 68, 3236. With this, as I noticed here, we finished reproducing all main results from that paper.
  • Finally we looked here at the proyection of the "electronic" state we found earlier into its phonon coordinates and pointed out its similarities to the ground state.

We also continued revising more literature and I'm trying to reflect the main concepts I'm learning in my Research Wiki. In a couple of days I will make a front page for all my subdomains and proyects in course.

Next Page »