Confessions of a Diagonalcoholic!

Every Quantum-problem-solver person is a repeat-offender of a simple crime! Diagonalizing!

I am writing about the philosophy of diagonalizing in a bid to elucidate the underlying motive of the QuTiP functions I am writing.

I am writing about a humble problem of calculating the dispersion relationship. Although, algorithmically or computationally it is not intimidating, it exemplifies one of the most effective diagonalization technique at our disposal-- translational symmetry!

In exact diagonalization of an interacting model translational symmetry technique can become the difference between an in and out-of-reach problem. If a 1d lattice can be resolved into a unit cell repeating N times, the translational symmetry can help reduce the length of the matrix to be diagonalized by a factor of N. For interacting models which scale as 2(or 4) to the power N, this little reduction in size can make a problem solvable with a given computational resource.

Using other symmetries can help with further dimensional reduction. I shall elaborate on our work on a Fermi Hubbard model soon.


Comments

Post a Comment

Popular posts from this blog

What I cannot code-up, I do not understand

There you go! Paraphrasing Richard Feynman-- done! Google Summer of Code -- done! I have gained valuable experience from the internship. Also, I hope to become a part of QuTiP soon. The title of this post more or less sums up my experience from the program. This will be a poor man's version of Feynman's dying quotes(on his blackboard!) "What I cannot create, I do not understand" "Know how to solve every problem that has been solved." With some awesome guidance from my mentors, I have been able to simulate some very well known concepts of lattice physics. Recreating a problem gives me an immense satisfaction as well as the confidence of understanding a problem inside out. I have never been in the elite bracket of physicists who can solve any problem on a piece of paper or a board. I am a computational person. So, to me redoing a problem means coding it up on a computer. I daresay, the list of problems that I believe to understand well has grown con...
Read more

The storm before the calm?

We are very close to the end! I need certain things to click. As I have come to learn, cracking the algorithmic part is a small part of the coding job. A code that produces accurate results have to withstand iterations of scrutiny, examination and cross-examination before it is given the green light. But, for me, the cracking part is the most crucial part. The project/functionality is made possible because an accurate code was produced, however 'unclean' and unscrutable. Everything hinges on that first breakthrough. I had to go over all the mathematics of Branch cuts in logarithmic functions to write a unfailing Chern number calculation routine. The most important functionality I want to add is the Chern number calculator function. This is not a simple calculation that can be done with any brute force way. It involves a intimate knowledge of branch cut of complex Logarithmic functions and also highlights the Gauge dependency of quantities in a Quantum problem. I am  havi...
Read more

A round peg in a round hole (Subject to Change!!)

At first, I did not think it necessary to add the "subject-to-change" clause at the end of the title, but now I know better! The process of first getting the correct functionality to actually convince the developers of QuTiP of the 'mergabilty'/readiness of the code is painstaking. Long before I managed to convince my mentors of the readiness of the code, I was able to get the codes to work and Writing codes for a popular simulation tool is much more demanding than writing codes for your research. My codes have already undergone a complete overhaul, almost everyday change, so as to make the code compatible with the spirit and soul of QuTiP. The science problem is only a fraction of the challenge. Finding the most convenient data structure and optimization philosophy is the . The code capability is crucially dependent on data structures and memory usage. I have had to spend a massive amount of time restructuring the code and developing documentation. Please ch...
Read more