---
_id: '7790'
abstract:
- lang: eng
text: "We prove a lower bound for the free energy (per unit volume) of the two-dimensional
Bose gas in the thermodynamic limit. We show that the free energy at density \U0001D70C
and inverse temperature \U0001D6FD differs from the one of the noninteracting
system by the correction term \U0001D70B\U0001D70C\U0001D70C\U0001D6FD\U0001D6FD
. Here, is the scattering length of the interaction potential, and \U0001D6FD
is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity.
The result is valid in the dilute limit \U0001D70C and if \U0001D6FD\U0001D70C
."
article_number: e20
article_processing_charge: No
article_type: original
author:
- first_name: Andreas
full_name: Deuchert, Andreas
id: 4DA65CD0-F248-11E8-B48F-1D18A9856A87
last_name: Deuchert
orcid: 0000-0003-3146-6746
- first_name: Simon
full_name: Mayer, Simon
id: 30C4630A-F248-11E8-B48F-1D18A9856A87
last_name: Mayer
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: Deuchert A, Mayer S, Seiringer R. The free energy of the two-dimensional dilute
Bose gas. I. Lower bound. Forum of Mathematics, Sigma. 2020;8. doi:10.1017/fms.2020.17
apa: Deuchert, A., Mayer, S., & Seiringer, R. (2020). The free energy of the
two-dimensional dilute Bose gas. I. Lower bound. Forum of Mathematics, Sigma.
Cambridge University Press. https://doi.org/10.1017/fms.2020.17
chicago: Deuchert, Andreas, Simon Mayer, and Robert Seiringer. “The Free Energy
of the Two-Dimensional Dilute Bose Gas. I. Lower Bound.” Forum of Mathematics,
Sigma. Cambridge University Press, 2020. https://doi.org/10.1017/fms.2020.17.
ieee: A. Deuchert, S. Mayer, and R. Seiringer, “The free energy of the two-dimensional
dilute Bose gas. I. Lower bound,” Forum of Mathematics, Sigma, vol. 8.
Cambridge University Press, 2020.
ista: Deuchert A, Mayer S, Seiringer R. 2020. The free energy of the two-dimensional
dilute Bose gas. I. Lower bound. Forum of Mathematics, Sigma. 8, e20.
mla: Deuchert, Andreas, et al. “The Free Energy of the Two-Dimensional Dilute Bose
Gas. I. Lower Bound.” Forum of Mathematics, Sigma, vol. 8, e20, Cambridge
University Press, 2020, doi:10.1017/fms.2020.17.
short: A. Deuchert, S. Mayer, R. Seiringer, Forum of Mathematics, Sigma 8 (2020).
date_created: 2020-05-03T22:00:48Z
date_published: 2020-03-14T00:00:00Z
date_updated: 2023-08-21T06:18:49Z
day: '14'
ddc:
- '510'
department:
- _id: RoSe
doi: 10.1017/fms.2020.17
ec_funded: 1
external_id:
arxiv:
- '1910.03372'
isi:
- '000527342000001'
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file_size: 692530
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month: '03'
oa: 1
oa_version: Published Version
project:
- _id: 25C6DC12-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '694227'
name: Analysis of quantum many-body systems
publication: Forum of Mathematics, Sigma
publication_identifier:
eissn:
- '20505094'
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
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scopus_import: '1'
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title: The free energy of the two-dimensional dilute Bose gas. I. Lower bound
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: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 8
year: '2020'
...
---
_id: '8134'
abstract:
- lang: eng
text: We prove an upper bound on the free energy of a two-dimensional homogeneous
Bose gas in the thermodynamic limit. We show that for a2ρ ≪ 1 and βρ ≳ 1, the
free energy per unit volume differs from the one of the non-interacting system
by at most 4πρ2|lna2ρ|−1(2−[1−βc/β]2+) to leading order, where a is the scattering
length of the two-body interaction potential, ρ is the density, β is the inverse
temperature, and βc is the inverse Berezinskii–Kosterlitz–Thouless critical temperature
for superfluidity. In combination with the corresponding matching lower bound
proved by Deuchert et al. [Forum Math. Sigma 8, e20 (2020)], this shows equality
in the asymptotic expansion.
article_number: '061901'
article_processing_charge: No
article_type: original
author:
- first_name: Simon
full_name: Mayer, Simon
id: 30C4630A-F248-11E8-B48F-1D18A9856A87
last_name: Mayer
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: Mayer S, Seiringer R. The free energy of the two-dimensional dilute Bose gas.
II. Upper bound. Journal of Mathematical Physics. 2020;61(6). doi:10.1063/5.0005950
apa: Mayer, S., & Seiringer, R. (2020). The free energy of the two-dimensional
dilute Bose gas. II. Upper bound. Journal of Mathematical Physics. AIP
Publishing. https://doi.org/10.1063/5.0005950
chicago: Mayer, Simon, and Robert Seiringer. “The Free Energy of the Two-Dimensional
Dilute Bose Gas. II. Upper Bound.” Journal of Mathematical Physics. AIP
Publishing, 2020. https://doi.org/10.1063/5.0005950.
ieee: S. Mayer and R. Seiringer, “The free energy of the two-dimensional dilute
Bose gas. II. Upper bound,” Journal of Mathematical Physics, vol. 61, no.
6. AIP Publishing, 2020.
ista: Mayer S, Seiringer R. 2020. The free energy of the two-dimensional dilute
Bose gas. II. Upper bound. Journal of Mathematical Physics. 61(6), 061901.
mla: Mayer, Simon, and Robert Seiringer. “The Free Energy of the Two-Dimensional
Dilute Bose Gas. II. Upper Bound.” Journal of Mathematical Physics, vol.
61, no. 6, 061901, AIP Publishing, 2020, doi:10.1063/5.0005950.
short: S. Mayer, R. Seiringer, Journal of Mathematical Physics 61 (2020).
date_created: 2020-07-19T22:00:59Z
date_published: 2020-06-22T00:00:00Z
date_updated: 2023-08-22T08:12:40Z
day: '22'
department:
- _id: RoSe
doi: 10.1063/5.0005950
ec_funded: 1
external_id:
arxiv:
- '2002.08281'
isi:
- '000544595100001'
intvolume: ' 61'
isi: 1
issue: '6'
language:
- iso: eng
main_file_link:
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url: https://arxiv.org/abs/2002.08281
month: '06'
oa: 1
oa_version: Preprint
project:
- _id: 25C6DC12-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '694227'
name: Analysis of quantum many-body systems
publication: Journal of Mathematical Physics
publication_identifier:
issn:
- '00222488'
publication_status: published
publisher: AIP Publishing
quality_controlled: '1'
scopus_import: '1'
status: public
title: The free energy of the two-dimensional dilute Bose gas. II. Upper bound
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 61
year: '2020'
...
---
_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:
- ISTA 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.
Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:7514
chicago: Mayer, Simon. “The Free Energy of a Dilute Two-Dimensional Bose Gas.” Institute
of Science and Technology Austria, 2020. https://doi.org/10.15479/AT:ISTA:7514.
ieee: S. Mayer, “The free energy of a dilute two-dimensional Bose gas,” Institute
of Science and Technology Austria, 2020.
ista: Mayer S. 2020. The free energy of a dilute two-dimensional Bose gas. Institute
of Science and Technology Austria.
mla: Mayer, Simon. The Free Energy of a Dilute Two-Dimensional Bose Gas.
Institute of Science and Technology Austria, 2020, doi:10.15479/AT:ISTA:7514.
short: S. Mayer, The Free Energy of a Dilute Two-Dimensional Bose Gas, Institute
of Science and Technology Austria, 2020.
date_created: 2020-02-24T09:17:27Z
date_published: 2020-02-24T00:00:00Z
date_updated: 2023-09-07T13:12:42Z
day: '24'
ddc:
- '510'
degree_awarded: PhD
department:
- _id: RoSe
- _id: GradSch
doi: 10.15479/AT:ISTA:7514
ec_funded: 1
file:
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checksum: b4de7579ddc1dbdd44ff3f17c48395f6
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creator: dernst
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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: Institute of Science and Technology Austria
related_material:
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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: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2020'
...
---
_id: '7524'
abstract:
- lang: eng
text: "We prove a lower bound for the free energy (per unit volume) of the two-dimensional
Bose gas in the thermodynamic limit. We show that the free energy at density $\\rho$
and inverse temperature $\\beta$ differs from the one of the non-interacting system
by the correction term $4 \\pi \\rho^2 |\\ln a^2 \\rho|^{-1} (2 - [1 - \\beta_{\\mathrm{c}}/\\beta]_+^2)$.
Here $a$ is the scattering length of the interaction potential, $[\\cdot]_+ =
\\max\\{ 0, \\cdot \\}$ and $\\beta_{\\mathrm{c}}$ is the inverse Berezinskii--Kosterlitz--Thouless
critical temperature for superfluidity. The result is valid in the dilute limit\r\n$a^2\\rho
\\ll 1$ and if $\\beta \\rho \\gtrsim 1$."
article_processing_charge: No
author:
- first_name: Andreas
full_name: Deuchert, Andreas
id: 4DA65CD0-F248-11E8-B48F-1D18A9856A87
last_name: Deuchert
orcid: 0000-0003-3146-6746
- first_name: Simon
full_name: Mayer, Simon
id: 30C4630A-F248-11E8-B48F-1D18A9856A87
last_name: Mayer
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: Deuchert A, Mayer S, Seiringer R. The free energy of the two-dimensional dilute
Bose gas. I. Lower bound. arXiv:191003372.
apa: Deuchert, A., Mayer, S., & Seiringer, R. (n.d.). The free energy of the
two-dimensional dilute Bose gas. I. Lower bound. arXiv:1910.03372. ArXiv.
chicago: Deuchert, Andreas, Simon Mayer, and Robert Seiringer. “The Free Energy
of the Two-Dimensional Dilute Bose Gas. I. Lower Bound.” ArXiv:1910.03372.
ArXiv, n.d.
ieee: A. Deuchert, S. Mayer, and R. Seiringer, “The free energy of the two-dimensional
dilute Bose gas. I. Lower bound,” arXiv:1910.03372. ArXiv.
ista: Deuchert A, Mayer S, Seiringer R. The free energy of the two-dimensional dilute
Bose gas. I. Lower bound. arXiv:1910.03372, .
mla: Deuchert, Andreas, et al. “The Free Energy of the Two-Dimensional Dilute Bose
Gas. I. Lower Bound.” ArXiv:1910.03372, ArXiv.
short: A. Deuchert, S. Mayer, R. Seiringer, ArXiv:1910.03372 (n.d.).
date_created: 2020-02-26T08:46:40Z
date_published: 2019-10-08T00:00:00Z
date_updated: 2023-09-07T13:12:41Z
day: '08'
department:
- _id: RoSe
ec_funded: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1910.03372
month: '10'
oa: 1
oa_version: Preprint
page: '61'
project:
- _id: 25C6DC12-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '694227'
name: Analysis of quantum many-body systems
publication: arXiv:1910.03372
publication_status: draft
publisher: ArXiv
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title: The free energy of the two-dimensional dilute Bose gas. I. Lower bound
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2019'
...