---
_id: '7514'
abstract:
- lang: eng
text: "We study the interacting homogeneous Bose gas in two spatial dimensions in
the thermodynamic limit at fixed density. We shall be concerned with some mathematical
aspects of this complicated problem in many-body quantum mechanics. More specifically,
we consider the dilute limit where the scattering length of the interaction potential,
which is a measure for the effective range of the potential, is small compared
to the average distance between the particles. We are interested in a setting
with positive (i.e., non-zero) temperature. After giving a survey of the relevant
literature in the field, we provide some facts and examples to set expectations
for the two-dimensional system. The crucial difference to the three-dimensional
system is that there is no Bose–Einstein condensate at positive temperature due
to the Hohenberg–Mermin–Wagner theorem. However, it turns out that an asymptotic
formula for the free energy holds similarly to the three-dimensional case.\r\nWe
motivate this formula by considering a toy model with δ interaction potential.
By restricting this model Hamiltonian to certain trial states with a quasi-condensate
we obtain an upper bound for the free energy that still has the quasi-condensate
fraction as a free parameter. When minimizing over the quasi-condensate fraction,
we obtain the Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity,
which plays an important role in our rigorous contribution. The mathematically
rigorous result that we prove concerns the specific free energy in the dilute
limit. We give upper and lower bounds on the free energy in terms of the free
energy of the non-interacting system and a correction term coming from the interaction.
Both bounds match and thus we obtain the leading term of an asymptotic approximation
in the dilute limit, provided the thermal wavelength of the particles is of the
same order (or larger) than the average distance between the particles. The remarkable
feature of this result is its generality: the correction term depends on the interaction
potential only through its scattering length and it holds for all nonnegative
interaction potentials with finite scattering length that are measurable. In particular,
this allows to model an interaction of hard disks."
alternative_title:
- IST Austria Thesis
article_processing_charge: No
author:
- first_name: Simon
full_name: Mayer, Simon
id: 30C4630A-F248-11E8-B48F-1D18A9856A87
last_name: Mayer
citation:
ama: Mayer S. The free energy of a dilute two-dimensional Bose gas. 2020. doi:10.15479/AT:ISTA:7514
apa: Mayer, S. (2020). *The free energy of a dilute two-dimensional Bose gas*.
IST Austria. https://doi.org/10.15479/AT:ISTA:7514
chicago: Mayer, Simon. “The Free Energy of a Dilute Two-Dimensional Bose Gas.” IST
Austria, 2020. https://doi.org/10.15479/AT:ISTA:7514.
ieee: S. Mayer, “The free energy of a dilute two-dimensional Bose gas,” IST Austria,
2020.
ista: Mayer S. 2020. The free energy of a dilute two-dimensional Bose gas. IST Austria.
mla: Mayer, Simon. *The Free Energy of a Dilute Two-Dimensional Bose Gas*.
IST Austria, 2020, doi:10.15479/AT:ISTA:7514.
short: S. Mayer, The Free Energy of a Dilute Two-Dimensional Bose Gas, IST Austria,
2020.
date_created: 2020-02-24T09:17:27Z
date_published: 2020-02-24T00:00:00Z
date_updated: 2021-01-12T08:14:01Z
day: '24'
ddc:
- '510'
department:
- _id: RoSe
doi: 10.15479/AT:ISTA:7514
ec_funded: 1
file:
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date_created: 2020-02-24T09:15:06Z
date_updated: 2020-07-14T12:47:59Z
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file_name: thesis.pdf
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license: https://creativecommons.org/licenses/by/4.0/
month: '02'
oa: 1
oa_version: Published Version
page: '148'
project:
- _id: 25C6DC12-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '694227'
name: Analysis of quantum many-body systems
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: IST Austria
related_material:
record:
- id: '7524'
relation: part_of_dissertation
status: public
status: public
supervisor:
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
title: The free energy of a dilute two-dimensional Bose gas
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: dissertation
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2020'
...
---
_id: '52'
abstract:
- lang: eng
text: In this thesis we will discuss systems of point interacting fermions, their
stability and other spectral properties. Whereas for bosons a point interacting
system is always unstable this ques- tion is more subtle for a gas of two species
of fermions. In particular the answer depends on the mass ratio between these
two species. Most of this work will be focused on the N + M model which consists
of two species of fermions with N, M particles respectively which interact via
point interactions. We will introduce this model using a formal limit and discuss
the N + 1 system in more detail. In particular, we will show that for mass ratios
above a critical one, which does not depend on the particle number, the N + 1
system is stable. In the context of this model we will prove rigorous versions
of Tan relations which relate various quantities of the point-interacting model.
By restricting the N + 1 system to a box we define a finite density model with
point in- teractions. In the context of this system we will discuss the energy
change when introducing a point-interacting impurity into a system of non-interacting
fermions. We will see that this change in energy is bounded independently of the
particle number and in particular the bound only depends on the density and the
scattering length. As another special case of the N + M model we will show stability
of the 2 + 2 model for mass ratios in an interval around one. Further we will
investigate a different model of point interactions which was discussed before
in the literature and which is, contrary to the N + M model, not given by a limiting
procedure but is based on a Dirichlet form. We will show that this system behaves
trivially in the thermodynamic limit, i.e. the free energy per particle is the
same as the one of the non-interacting system.
alternative_title:
- IST Austria Thesis
author:
- first_name: Thomas
full_name: Moser, Thomas
id: 2B5FC9A4-F248-11E8-B48F-1D18A9856A87
last_name: Moser
citation:
ama: Moser T. Point interactions in systems of fermions. 2018. doi:10.15479/AT:ISTA:th_1043
apa: Moser, T. (2018). *Point interactions in systems of fermions*. IST Austria.
https://doi.org/10.15479/AT:ISTA:th_1043
chicago: Moser, Thomas. “Point Interactions in Systems of Fermions.” IST Austria,
2018. https://doi.org/10.15479/AT:ISTA:th_1043.
ieee: T. Moser, “Point interactions in systems of fermions,” IST Austria, 2018.
ista: Moser T. 2018. Point interactions in systems of fermions. IST Austria.
mla: Moser, Thomas. *Point Interactions in Systems of Fermions*. IST Austria,
2018, doi:10.15479/AT:ISTA:th_1043.
short: T. Moser, Point Interactions in Systems of Fermions, IST Austria, 2018.
date_created: 2018-12-11T11:44:22Z
date_published: 2018-09-04T00:00:00Z
date_updated: 2021-01-12T08:13:26Z
day: '04'
ddc:
- '515'
- '530'
- '519'
department:
- _id: RoSe
doi: 10.15479/AT:ISTA:th_1043
file:
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checksum: c28e16ecfc1126d3ce324ec96493c01e
content_type: application/zip
creator: dernst
date_created: 2019-04-09T07:45:38Z
date_updated: 2020-07-14T12:46:37Z
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file_name: 2018_Thesis_Moser_Source.zip
file_size: 1531516
relation: source_file
file_date_updated: 2020-07-14T12:46:37Z
has_accepted_license: '1'
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
page: '115'
project:
- _id: 25C878CE-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P27533_N27
name: Structure of the Excitation Spectrum for Many-Body Quantum Systems
publication_status: published
publisher: IST Austria
publist_id: '8002'
pubrep_id: '1043'
related_material:
record:
- id: '1198'
relation: part_of_dissertation
status: public
- id: '154'
relation: part_of_dissertation
status: public
- id: '5856'
relation: part_of_dissertation
status: public
- id: '741'
relation: part_of_dissertation
status: public
status: public
supervisor:
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
title: Point interactions in systems of fermions
type: dissertation
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2018'
...