Flow equations for cold Bose gases
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| Number of Parts | 40 | |
| Author | 0000-0003-0393-5525 (ORCID) 0000-0002-2318-0644 (ORCID) | |
| License | CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. | |
| Identifiers | 10.5446/38453 (DOI) | |
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VideoPaper
Transcript: English(auto-generated)
00:03
Hello, my name is Artin. I am happy to introduce to you our paper, Flow Equations for Cold Bose Gases. In quantum mechanics we often deal with matrices. We say that the problem of interest is solved when the corresponding matrix is diagonalized.
00:21
The diagonalization of a matrix can be achieved using a unitary operator U. To illustrate the action of U, let us use two curves. A real one which is a complicated object, and a transformed spherical curve that dream of a theoretical physicist because it can be easily analyzed.
00:43
The first curve here represents the initial matrix. The second is the diagonal matrix with the same information content. It is a complicated task to find U because it connects two objects that look very different. Instead of calculating U,
01:01
one can diagonalize the matrix step by step using a sequence of small unitary transformations. This approach is described by a set of differential equations often referred to as the flow equations. A derivation of such a set for cold Bose gases is the main objective of our paper.
01:23
In order to write the flow equations, we should define the generator eta. The evolution should eliminate all of the diagonal terms of the Hamiltonian. Therefore, a reasonably simple choice of eta has to contain one and two body terms.
01:41
However, such a choice leads to the appearance of n body forces during the evolution. It is an impossible task to evolve exactly all n body terms, and thus we have to truncate the Hamiltonian in order to calculate the energies. Through this end, we introduce a reference state to normal order operators,
02:03
and then we truncate the Hamiltonian at the level of three body forces and beyond. To estimate the accuracy of our results and to calculate corrections, we treat the neglected pieces as perturbation. To test our approach, we use the Leibniger guess. It is an exactly solvable model
02:21
that describes n bosons in one dimension. Using flow equations, we calculate the ground state energies of the guess for n equals 4 in 15. We show that our method reproduces the exact results for weak and moderate interactions. We also show that the suggesting procedure
02:40
for estimating errors works well. In the future, we would like to use a developed method to study two- and three-dimensional systems that cannot be solved analytically. Also, we would like to study systems with impurities. We have already performed some calculations using the flow equations for the lattice system.
03:02
We calculated energies of an impenetrable impurity in a weakly interacting Bode guess. Thank you for watching this video. We hope that we motivated you to read our paper. Please do not hesitate to contact us if you have questions.