---
_id: '14374'
abstract:
- lang: eng
text: "Superconductivity has many important applications ranging from levitating
trains over qubits to MRI scanners. The phenomenon is successfully modeled by
Bardeen-Cooper-Schrieffer (BCS) theory. From a mathematical perspective, BCS theory
has been studied extensively for systems without boundary. However, little is
known in the presence of boundaries. With the help of numerical methods physicists
observed that the critical temperature may increase in the presence of a boundary.
The goal of this thesis is to understand the influence of boundaries on the critical
temperature in BCS theory and to give a first rigorous justification of these
observations. On the way, we also study two-body Schrödinger operators on domains
with boundaries and prove additional results for superconductors without boundary.\r\n\r\nBCS
theory is based on a non-linear functional, where the minimizer indicates whether
the system is superconducting or in the normal, non-superconducting state. By
considering the Hessian of the BCS functional at the normal state, one can analyze
whether the normal state is possibly a minimum of the BCS functional and estimate
the critical temperature. The Hessian turns out to be a linear operator resembling
a Schrödinger operator for two interacting particles, but with more complicated
kinetic energy. As a first step, we study the two-body Schrödinger operator in
the presence of boundaries.\r\nFor Neumann boundary conditions, we prove that
the addition of a boundary can create new eigenvalues, which correspond to the
two particles forming a bound state close to the boundary.\r\n\r\nSecond, we need
to understand superconductivity in the translation invariant setting. While in
three dimensions this has been extensively studied, there is no mathematical literature
for the one and two dimensional cases. In dimensions one and two, we compute the
weak coupling asymptotics of the critical temperature and the energy gap in the
translation invariant setting. We also prove that their ratio is independent of
the microscopic details of the model in the weak coupling limit; this property
is referred to as universality.\r\n\r\nIn the third part, we study the critical
temperature of superconductors in the presence of boundaries. We start by considering
the one-dimensional case of a half-line with contact interaction. Then, we generalize
the results to generic interactions and half-spaces in one, two and three dimensions.
Finally, we compare the critical temperature of a quarter space in two dimensions
to the critical temperatures of a half-space and of the full space."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Barbara
full_name: Roos, Barbara
id: 5DA90512-D80F-11E9-8994-2E2EE6697425
last_name: Roos
orcid: 0000-0002-9071-5880
citation:
ama: Roos B. Boundary superconductivity in BCS theory. 2023. doi:10.15479/at:ista:14374
apa: Roos, B. (2023). Boundary superconductivity in BCS theory. Institute
of Science and Technology Austria. https://doi.org/10.15479/at:ista:14374
chicago: Roos, Barbara. “Boundary Superconductivity in BCS Theory.” Institute of
Science and Technology Austria, 2023. https://doi.org/10.15479/at:ista:14374.
ieee: B. Roos, “Boundary superconductivity in BCS theory,” Institute of Science
and Technology Austria, 2023.
ista: Roos B. 2023. Boundary superconductivity in BCS theory. Institute of Science
and Technology Austria.
mla: Roos, Barbara. Boundary Superconductivity in BCS Theory. Institute of
Science and Technology Austria, 2023, doi:10.15479/at:ista:14374.
short: B. Roos, Boundary Superconductivity in BCS Theory, Institute of Science and
Technology Austria, 2023.
date_created: 2023-09-28T14:23:04Z
date_published: 2023-09-30T00:00:00Z
date_updated: 2023-10-27T10:37:30Z
day: '30'
ddc:
- '515'
- '539'
degree_awarded: PhD
department:
- _id: GradSch
- _id: RoSe
doi: 10.15479/at:ista:14374
ec_funded: 1
file:
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language:
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license: https://creativecommons.org/licenses/by-nc-sa/4.0/
month: '09'
oa: 1
oa_version: Published Version
page: '206'
project:
- _id: 25C6DC12-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '694227'
name: Analysis of quantum many-body systems
- _id: bda63fe5-d553-11ed-ba76-a16e3d2f256b
grant_number: I06427
name: Mathematical Challenges in BCS Theory of Superconductivity
publication_identifier:
issn:
- 2663 - 337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
record:
- id: '13207'
relation: part_of_dissertation
status: public
- id: '10850'
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: Boundary superconductivity in BCS theory
tmp:
image: /images/cc_by_nc_sa.png
legal_code_url: https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode
name: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC
BY-NC-SA 4.0)
short: CC BY-NC-SA (4.0)
type: dissertation
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2023'
...
---
_id: '12390'
abstract:
- lang: eng
text: "The scope of this thesis is to study quantum systems exhibiting a continuous
symmetry that\r\nis broken on the level of the corresponding effective theory.
In particular we are going to\r\ninvestigate translation-invariant Bose gases
in the mean field limit, effectively described by\r\nthe Hartree functional, and
the Fröhlich Polaron in the regime of strong coupling, effectively\r\ndescribed
by the Pekar functional. The latter is a model describing the interaction between
a\r\ncharged particle and the optical modes of a polar crystal. Regarding the
former, we assume in\r\naddition that the particles in the gas are unconfined,
and typically we will consider particles\r\nthat are subject to an attractive
interaction. In both cases the ground state energy of the\r\nHamiltonian is not
a proper eigenvalue due to the underlying translation-invariance, while on\r\nthe
contrary there exists a whole invariant orbit of minimizers for the corresponding
effective\r\nfunctionals. Both, the absence of proper eigenstates and the broken
symmetry of the effective\r\ntheory, make the study significantly more involved
and it is the content of this thesis to\r\ndevelop a frameworks which allows for
a systematic way to circumvent these issues.\r\nIt is a well-established result
that the ground state energy of Bose gases in the mean field limit,\r\nas well
as the ground state energy of the Fröhlich Polaron in the regime of strong coupling,
is\r\nto leading order given by the minimal energy of the corresponding effective
theory. As part\r\nof this thesis we identify the sub-leading term in the expansion
of the ground state energy,\r\nwhich can be interpreted as the quantum correction
to the classical energy, since the effective\r\ntheories under consideration can
be seen as classical counterparts.\r\nWe are further going to establish an asymptotic
expression for the energy-momentum relation\r\nof the Fröhlich Polaron in the
strong coupling limit. In the regime of suitably small momenta,\r\nthis asymptotic
expression agrees with the energy-momentum relation of a free particle having\r\nan
effectively increased mass, and we find that this effectively increased mass agrees
with the\r\nconjectured value in the physics literature.\r\nIn addition we will
discuss two unrelated papers written by the author during his stay at ISTA\r\nin
the appendix. The first one concerns the realization of anyons, which are quasi-particles\r\nacquiring
a non-trivial phase under the exchange of two particles, as molecular impurities.\r\nThe
second one provides a classification of those vector fields defined on a given
manifold\r\nthat can be written as the gradient of a given functional with respect
to a suitable metric,\r\nprovided that some mild smoothness assumptions hold.
This classification is subsequently\r\nused to identify those quantum Markov semigroups
that can be written as a gradient flow of\r\nthe relative entropy.\r\n"
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Morris
full_name: Brooks, Morris
id: B7ECF9FC-AA38-11E9-AC9A-0930E6697425
last_name: Brooks
orcid: 0000-0002-6249-0928
citation:
ama: Brooks M. Translation-invariant quantum systems with effectively broken symmetry.
2022. doi:10.15479/at:ista:12390
apa: Brooks, M. (2022). Translation-invariant quantum systems with effectively
broken symmetry. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:12390
chicago: Brooks, Morris. “Translation-Invariant Quantum Systems with Effectively
Broken Symmetry.” Institute of Science and Technology Austria, 2022. https://doi.org/10.15479/at:ista:12390.
ieee: M. Brooks, “Translation-invariant quantum systems with effectively broken
symmetry,” Institute of Science and Technology Austria, 2022.
ista: Brooks M. 2022. Translation-invariant quantum systems with effectively broken
symmetry. Institute of Science and Technology Austria.
mla: Brooks, Morris. Translation-Invariant Quantum Systems with Effectively Broken
Symmetry. Institute of Science and Technology Austria, 2022, doi:10.15479/at:ista:12390.
short: M. Brooks, Translation-Invariant Quantum Systems with Effectively Broken
Symmetry, Institute of Science and Technology Austria, 2022.
date_created: 2023-01-26T10:00:42Z
date_published: 2022-12-15T00:00:00Z
date_updated: 2023-08-07T13:32:09Z
day: '15'
ddc:
- '500'
degree_awarded: PhD
department:
- _id: GradSch
- _id: RoSe
doi: 10.15479/at:ista:12390
ec_funded: 1
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checksum: b31460e937f33b557abb40ebef02b567
content_type: application/pdf
creator: cchlebak
date_created: 2023-01-26T10:02:34Z
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date_created: 2023-01-26T10:02:42Z
date_updated: 2023-01-26T10:02:42Z
file_id: '12392'
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file_date_updated: 2023-01-26T10:02:42Z
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language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: '196'
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:
record:
- id: '9005'
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: Translation-invariant quantum systems with effectively broken symmetry
tmp:
image: /images/cc_by_nc_sa.png
legal_code_url: https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode
name: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC
BY-NC-SA 4.0)
short: CC BY-NC-SA (4.0)
type: dissertation
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2022'
...
---
_id: '11473'
abstract:
- lang: eng
text: "The polaron model is a basic model of quantum field theory describing a single
particle\r\ninteracting with a bosonic field. It arises in many physical contexts.
We are mostly concerned\r\nwith models applicable in the context of an impurity
atom in a Bose-Einstein condensate as\r\nwell as the problem of electrons moving
in polar crystals.\r\nThe model has a simple structure in which the interaction
of the particle with the field is given\r\nby a term linear in the field’s creation
and annihilation operators. In this work, we investigate\r\nthe properties of
this model by providing rigorous estimates on various energies relevant to the\r\nproblem.
The estimates are obtained, for the most part, by suitable operator techniques
which\r\nconstitute the principal mathematical substance of the thesis.\r\nThe
first application of these techniques is to derive the polaron model rigorously
from first\r\nprinciples, i.e., from a full microscopic quantum-mechanical many-body
problem involving an\r\nimpurity in an otherwise homogeneous system. We accomplish
this for the N + 1 Bose gas\r\nin the mean-field regime by showing that a suitable
polaron-type Hamiltonian arises at weak\r\ninteractions as a low-energy effective
theory for this problem.\r\nIn the second part, we investigate rigorously the
ground state of the model at fixed momentum\r\nand for large values of the coupling
constant. Qualitatively, the system is expected to display\r\na transition from
the quasi-particle behavior at small momenta, where the dispersion relation\r\nis
parabolic and the particle moves through the medium dragging along a cloud of
phonons, to\r\nthe radiative behavior at larger momenta where the polaron decelerates
and emits free phonons.\r\nAt the same time, in the strong coupling regime, the
bosonic field is expected to behave purely\r\nclassically. Accordingly, the effective
mass of the polaron at strong coupling is conjectured to\r\nbe asymptotically
equal to the one obtained from the semiclassical counterpart of the problem,\r\nfirst
studied by Landau and Pekar in the 1940s. For polaron models with regularized
form\r\nfactors and phonon dispersion relations of superfluid type, i.e., bounded
below by a linear\r\nfunction of the wavenumbers for all phonon momenta as in
the interacting Bose gas, we prove\r\nthat for a large window of momenta below
the radiation threshold, the energy-momentum\r\nrelation at strong coupling is
indeed essentially a parabola with semi-latus rectum equal to the\r\nLandau–Pekar
effective mass, as expected.\r\nFor the Fröhlich polaron describing electrons
in polar crystals where the dispersion relation is\r\nof the optical type and
the form factor is formally UV–singular due to the nature of the point\r\ncharge-dipole
interaction, we are able to give the corresponding upper bound. In contrast to\r\nthe
regular case, this requires the inclusion of the quantum fluctuations of the phonon
field,\r\nwhich makes the problem considerably more difficult.\r\nThe results
are supplemented by studies on the absolute ground-state energy at strong coupling,\r\na
proof of the divergence of the effective mass with the coupling constant for a
wide class of\r\npolaron models, as well as the discussion of the apparent UV
singularity of the Fröhlich model\r\nand the application of the techniques used
for its removal for the energy estimates.\r\n"
acknowledged_ssus:
- _id: SSU
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Krzysztof
full_name: Mysliwy, Krzysztof
id: 316457FC-F248-11E8-B48F-1D18A9856A87
last_name: Mysliwy
citation:
ama: 'Mysliwy K. Polarons in Bose gases and polar crystals: Some rigorous energy
estimates. 2022. doi:10.15479/at:ista:11473'
apa: 'Mysliwy, K. (2022). Polarons in Bose gases and polar crystals: Some rigorous
energy estimates. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:11473'
chicago: 'Mysliwy, Krzysztof. “Polarons in Bose Gases and Polar Crystals: Some Rigorous
Energy Estimates.” Institute of Science and Technology Austria, 2022. https://doi.org/10.15479/at:ista:11473.'
ieee: 'K. Mysliwy, “Polarons in Bose gases and polar crystals: Some rigorous energy
estimates,” Institute of Science and Technology Austria, 2022.'
ista: 'Mysliwy K. 2022. Polarons in Bose gases and polar crystals: Some rigorous
energy estimates. Institute of Science and Technology Austria.'
mla: 'Mysliwy, Krzysztof. Polarons in Bose Gases and Polar Crystals: Some Rigorous
Energy Estimates. Institute of Science and Technology Austria, 2022, doi:10.15479/at:ista:11473.'
short: 'K. Mysliwy, Polarons in Bose Gases and Polar Crystals: Some Rigorous Energy
Estimates, Institute of Science and Technology Austria, 2022.'
date_created: 2022-06-30T12:15:03Z
date_published: 2022-07-01T00:00:00Z
date_updated: 2023-09-07T13:43:52Z
day: '01'
ddc:
- '515'
- '539'
degree_awarded: PhD
department:
- _id: GradSch
- _id: RoSe
doi: 10.15479/at:ista:11473
ec_funded: 1
file:
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checksum: 7970714a20a6052f75fb27a6c3e9976e
content_type: application/pdf
creator: kmysliwy
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date_updated: 2022-07-05T08:12:56Z
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creator: kmysliwy
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date_updated: 2022-07-05T08:17:12Z
file_id: '11487'
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language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
page: '138'
project:
- _id: 2564DBCA-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '665385'
name: International IST Doctoral Program
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
record:
- id: '10564'
relation: part_of_dissertation
status: public
- id: '8705'
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: 'Polarons in Bose gases and polar crystals: Some rigorous energy estimates'
type: dissertation
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2022'
...
---
_id: '9733'
abstract:
- lang: eng
text: This thesis is the result of the research carried out by the author during
his PhD at IST Austria between 2017 and 2021. It mainly focuses on the Fröhlich
polaron model, specifically to its regime of strong coupling. This model, which
is rigorously introduced and discussed in the introduction, has been of great
interest in condensed matter physics and field theory for more than eighty years.
It is used to describe an electron interacting with the atoms of a solid material
(the strength of this interaction is modeled by the presence of a coupling constant
α in the Hamiltonian of the system). The particular regime examined here, which
is mathematically described by considering the limit α →∞, displays many interesting
features related to the emergence of classical behavior, which allows for a simplified
effective description of the system under analysis. The properties, the range
of validity and a quantitative analysis of the precision of such classical approximations
are the main object of the present work. We specify our investigation to the study
of the ground state energy of the system, its dynamics and its effective mass.
For each of these problems, we provide in the introduction an overview of the
previously known results and a detailed account of the original contributions
by the author.
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Dario
full_name: Feliciangeli, Dario
id: 41A639AA-F248-11E8-B48F-1D18A9856A87
last_name: Feliciangeli
orcid: 0000-0003-0754-8530
citation:
ama: Feliciangeli D. The polaron at strong coupling. 2021. doi:10.15479/at:ista:9733
apa: Feliciangeli, D. (2021). The polaron at strong coupling. Institute of
Science and Technology Austria. https://doi.org/10.15479/at:ista:9733
chicago: Feliciangeli, Dario. “The Polaron at Strong Coupling.” Institute of Science
and Technology Austria, 2021. https://doi.org/10.15479/at:ista:9733.
ieee: D. Feliciangeli, “The polaron at strong coupling,” Institute of Science and
Technology Austria, 2021.
ista: Feliciangeli D. 2021. The polaron at strong coupling. Institute of Science
and Technology Austria.
mla: Feliciangeli, Dario. The Polaron at Strong Coupling. Institute of Science
and Technology Austria, 2021, doi:10.15479/at:ista:9733.
short: D. Feliciangeli, The Polaron at Strong Coupling, Institute of Science and
Technology Austria, 2021.
date_created: 2021-07-27T15:48:30Z
date_published: 2021-08-20T00:00:00Z
date_updated: 2024-03-06T12:30:44Z
day: '20'
ddc:
- '515'
- '519'
- '539'
degree_awarded: PhD
department:
- _id: GradSch
- _id: RoSe
- _id: JaMa
doi: 10.15479/at:ista:9733
ec_funded: 1
file:
- access_level: open_access
checksum: e88bb8ca43948abe060eb2d2fa719881
content_type: application/pdf
creator: dfelicia
date_created: 2021-08-19T14:03:48Z
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language:
- iso: eng
month: '08'
oa: 1
oa_version: Published Version
page: '180'
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '716117'
name: Optimal Transport and Stochastic Dynamics
- _id: 25C6DC12-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '694227'
name: Analysis of quantum many-body systems
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
grant_number: F6504
name: Taming Complexity in Partial Differential Systems
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
record:
- id: '9787'
relation: part_of_dissertation
status: public
- id: '9792'
relation: part_of_dissertation
status: public
- id: '9225'
relation: part_of_dissertation
status: public
- id: '9781'
relation: part_of_dissertation
status: public
- id: '9791'
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
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
title: The polaron at strong coupling
tmp:
image: /image/cc_by_nd.png
legal_code_url: https://creativecommons.org/licenses/by-nd/4.0/legalcode
name: Creative Commons Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0)
short: CC BY-ND (4.0)
type: dissertation
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2021'
...
---
_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:
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full_name: Mayer, Simon
id: 30C4630A-F248-11E8-B48F-1D18A9856A87
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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:
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- _id: GradSch
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grant_number: '694227'
name: Analysis of quantum many-body systems
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publisher: Institute of Science and Technology Austria
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status: public
supervisor:
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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:
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short: CC BY (4.0)
type: dissertation
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...
---
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abstract:
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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:
- ISTA Thesis
article_processing_charge: No
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. Institute
of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:th_1043
chicago: Moser, Thomas. “Point Interactions in Systems of Fermions.” Institute of
Science and Technology Austria, 2018. https://doi.org/10.15479/AT:ISTA:th_1043.
ieee: T. Moser, “Point interactions in systems of fermions,” Institute of Science
and Technology Austria, 2018.
ista: Moser T. 2018. Point interactions in systems of fermions. Institute of Science
and Technology Austria.
mla: Moser, Thomas. Point Interactions in Systems of Fermions. Institute
of Science and Technology Austria, 2018, doi:10.15479/AT:ISTA:th_1043.
short: T. Moser, Point Interactions in Systems of Fermions, Institute of Science
and Technology Austria, 2018.
date_created: 2018-12-11T11:44:22Z
date_published: 2018-09-04T00:00:00Z
date_updated: 2023-09-27T12:34:14Z
day: '04'
ddc:
- '515'
- '530'
- '519'
degree_awarded: PhD
department:
- _id: RoSe
doi: 10.15479/AT:ISTA:th_1043
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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_identifier:
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
publist_id: '8002'
pubrep_id: '1043'
related_material:
record:
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relation: part_of_dissertation
status: public
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relation: part_of_dissertation
status: public
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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: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2018'
...