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A mean-field approach to attractive few-body Fermi gas

Piibeleht, Morten LU (2016) FYSM60 20161
Department of Physics
Mathematical Physics
Abstract
Quantum many-body systems can be studied theoretically from first principles using microscopic models, but the computational cost of numerically exact calculations increases exponentially with the number of particles and the complexity of the interaction, limiting the systems that can be analysed. The Hartree-Fock-Bogoliubov (HFB) and quasiparticle random phase approximation (QRPA) methods, heavily used in nuclear physics, rely on mean-field approximations but scale more favourably with increasing system size and complexity. In this work we apply these methods to the problem of a two-dimensional harmonically trapped Fermi gas with attractive interactions, with the goal of investigating the validity of the approximations in the few-body... (More)
Quantum many-body systems can be studied theoretically from first principles using microscopic models, but the computational cost of numerically exact calculations increases exponentially with the number of particles and the complexity of the interaction, limiting the systems that can be analysed. The Hartree-Fock-Bogoliubov (HFB) and quasiparticle random phase approximation (QRPA) methods, heavily used in nuclear physics, rely on mean-field approximations but scale more favourably with increasing system size and complexity. In this work we apply these methods to the problem of a two-dimensional harmonically trapped Fermi gas with attractive interactions, with the goal of investigating the validity of the approximations in the few-body limit.

We implement the approximate methods in code and then compare them with exact calculations. We find that the HFB method can account for pairing correlations that arise due to the attractive interaction, predicting well the ground state energy even at high interaction strengths. Also, the excitation spectra predicted by the QRPA method qualitatively match the true spectra, reproducing the expected subset of excitations. Finally, we see that perturbative corrections can be used to improve the ground state energy estimate of the HFB. (Less)
Popular Abstract
The past hundred or so years have shown us that if the microscopic world is to be described, quantum mechanics has to be invoked. It allows us to make accurate predictions on the scale of atoms and smaller, and at very low temperatures, down all the way to absolute zero. Whether it is a few tens of electrons whirling around in the electron cloud of an atom, or the billions of them navigating the crystal structure of piece of metal, we have the basic description down.

However, the difficulty with exact, first principles approaches is that the calculations can be computationally expensive. You could generally say that the cost originates from two sources: the size of the system, like the number of particles, and the complexity of the... (More)
The past hundred or so years have shown us that if the microscopic world is to be described, quantum mechanics has to be invoked. It allows us to make accurate predictions on the scale of atoms and smaller, and at very low temperatures, down all the way to absolute zero. Whether it is a few tens of electrons whirling around in the electron cloud of an atom, or the billions of them navigating the crystal structure of piece of metal, we have the basic description down.

However, the difficulty with exact, first principles approaches is that the calculations can be computationally expensive. You could generally say that the cost originates from two sources: the size of the system, like the number of particles, and the complexity of the system – the finer the details the system exhibits, the higher the accuracy needed for the calculations.

As an example, with several interacting particles, we have to account for the interaction between every pair. With more and more particles, the number of interacting pairs we have to keep track of will grow faster than the number of particles itself, and therefore the calculations can only be performed for relatively small systems. And by small we mean really small – going past a dozen particles is generally a major achievement. But then, how do we study these systems? The answer is “approximations”. What it means is that by using reasonable assumptions we reformulate and simplify the original problem, making the calculations easier. For example, instead of tracking every interaction between each pair of particles, we could instead approximate the interaction as an average effect from all the particles. That is, we say that every particle feels an average force from all the other particles, instead of being aware of them individually.

However, with approximation methods it can be difficult to know how well they work. In general, you could say that approximations always have a region of validity and outside of that region the method does not yield good results. Averaging the interaction would fail to describe the case where particles prefer to form pairs.

This work looks at a method that is basically averages out the interaction, but in a slightly more advanced way. It should also be able to account for the pairing between the particles, but the question we are asking is how well does it actually work. Our strategy? To compare the exact and approximate methods in the limit where it is possible to solve both. We will then find out how well it works and try to understand the source of any differences.

It turns out that the method can actually reproduce the relevant features reasonably well when we compare against the exact results. Some care is necessary when interpreting the results, but it seems that we have a tool at our disposal that we can use to probe larger systems than was possible before. (Less)
Please use this url to cite or link to this publication:
author
Piibeleht, Morten LU
supervisor
organization
course
FYSM60 20161
year
type
H2 - Master's Degree (Two Years)
subject
keywords
HFB, hartree-fock-bogoliubov, fermi gas, higgs mode, QRPA, quasiparticles, random phase approximation
language
English
id
8886518
date added to LUP
2016-07-04 11:26:00
date last changed
2016-11-15 13:52:21
@misc{8886518,
  abstract     = {Quantum many-body systems can be studied theoretically from first principles using microscopic models, but the computational cost of numerically exact calculations increases exponentially with the number of particles and the complexity of the interaction, limiting the systems that can be analysed. The Hartree-Fock-Bogoliubov (HFB) and quasiparticle random phase approximation (QRPA) methods, heavily used in nuclear physics, rely on mean-field approximations but scale more favourably with increasing system size and complexity. In this work we apply these methods to the problem of a two-dimensional harmonically trapped Fermi gas with attractive interactions, with the goal of investigating the validity of the approximations in the few-body limit.

We implement the approximate methods in code and then compare them with exact calculations. We find that the HFB method can account for pairing correlations that arise due to the attractive interaction, predicting well the ground state energy even at high interaction strengths. Also, the excitation spectra predicted by the QRPA method qualitatively match the true spectra, reproducing the expected subset of excitations. Finally, we see that perturbative corrections can be used to improve the ground state energy estimate of the HFB.},
  author       = {Piibeleht, Morten},
  keyword      = {HFB,hartree-fock-bogoliubov,fermi gas,higgs mode,QRPA,quasiparticles,random phase approximation},
  language     = {eng},
  note         = {Student Paper},
  title        = {A mean-field approach to attractive few-body Fermi gas},
  year         = {2016},
}