---
OA_place: publisher
_id: '19540'
abstract:
- lang: eng
text: "This thesis deals with several different models for complex quantum mechanical
systems and is structured in three main parts. \r\n\t\r\nIn Part I, we study mean
field random matrices as models for quantum Hamiltonians. Our focus lies on proving
concentration estimates for resolvents of random matrices, so-called local laws,
mostly in the setting of multiple resolvents. These estimates have profound consequences
for eigenvector overlaps and thermalization problems. More concretely, we obtain,
e.g., the optimal eigenstate thermalization hypothesis (ETH) uniformly in the
spectrum for Wigner matrices, an optimal lower bound on non-Hermitian eigenvector
overlaps, and prethermalization for deformed Wigner matrices.\tIn order to prove
our novel multi-resolvent local laws, we develop and devise two main methods,
the static Psi-method and the dynamical Zigzag strategy. \r\n\t\r\nIn Part II,
we study Bardeen-Cooper-Schrieffer (BCS) theory, the standard mean field microscopic
theory of superconductivity. We focus on asymptotic formulas for the characteristic
critical temperature and energy gap of a superconductor and prove universality
of their ratio in various physical regimes. Additionally, we investigate multi-band
superconductors and show that inter-band coupling effects can only enhance the
critical temperature. \r\n\t\r\nIn Part III, we study quantum lattice systems.
On the one hand, we show a strong version of the local-perturbations-perturb-locally
(LPPL) principle for the ground state of weakly interacting quantum spin systems
with a uniform on-site gap. On the other hand, we introduce a notion of a local
gap and rigorously justify response theory and the Kubo formula under the weakened
assumption of a local gap. \r\n\t\r\nAdditionally, we discuss two classes of problems
which do not fit into the three main parts of the thesis. These are deformational
rigidity of Liouville metrics on the torus and relativistic toy models of particle
creation via interior-boundary-conditions (IBCs). "
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Sven Joscha
full_name: Henheik, Sven Joscha
id: 31d731d7-d235-11ea-ad11-b50331c8d7fb
last_name: Henheik
orcid: 0000-0003-1106-327X
citation:
ama: 'Henheik SJ. Modeling complex quantum systems: Random matrices, BCS theory,
and quantum lattice systems. 2025. doi:10.15479/AT-ISTA-19540'
apa: 'Henheik, S. J. (2025). Modeling complex quantum systems: Random matrices,
BCS theory, and quantum lattice systems. Institute of Science and Technology
Austria. https://doi.org/10.15479/AT-ISTA-19540'
chicago: 'Henheik, Sven Joscha. “Modeling Complex Quantum Systems: Random Matrices,
BCS Theory, and Quantum Lattice Systems.” Institute of Science and Technology
Austria, 2025. https://doi.org/10.15479/AT-ISTA-19540.'
ieee: 'S. J. Henheik, “Modeling complex quantum systems: Random matrices, BCS theory,
and quantum lattice systems,” Institute of Science and Technology Austria, 2025.'
ista: 'Henheik SJ. 2025. Modeling complex quantum systems: Random matrices, BCS
theory, and quantum lattice systems. Institute of Science and Technology Austria.'
mla: 'Henheik, Sven Joscha. Modeling Complex Quantum Systems: Random Matrices,
BCS Theory, and Quantum Lattice Systems. Institute of Science and Technology
Austria, 2025, doi:10.15479/AT-ISTA-19540.'
short: 'S.J. Henheik, Modeling Complex Quantum Systems: Random Matrices, BCS Theory,
and Quantum Lattice Systems, Institute of Science and Technology Austria, 2025.'
corr_author: '1'
date_created: 2025-04-10T21:21:18Z
date_published: 2025-04-10T00:00:00Z
date_updated: 2026-07-29T13:18:17Z
day: '10'
ddc:
- '519'
degree_awarded: PhD
department:
- _id: GradSch
- _id: LaEr
doi: 10.15479/AT-ISTA-19540
doi_confirm: '1'
ec_funded: 1
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language:
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month: '04'
oa: 1
oa_version: Published Version
page: '720'
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
call_identifier: H2020
grant_number: '101020331'
name: Random matrices beyond Wigner-Dyson-Mehta
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issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
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supervisor:
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
title: 'Modeling complex quantum systems: Random matrices, BCS theory, and quantum
lattice systems'
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type: dissertation
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
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