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003			AT-ISTA
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040			_cISTA
100			_aHenheik, Sven Joscha _91084227
245			_aModeling complex quantum systems: Random matrices, BCS theory, and quantum lattice systems _bVolume 1 of 2
260			_bISTA _c2025
500			_aThesis
505			_aAbstract
505			_aAcknowledgements
505			_aAbout the Author
505			_aList of Collaborators and Publications
505			_aTable of Contents
505			_aList of Figures
505			_aList of Tables
505			_aIntroduction and Summary of Results
505			_aI Random Matrices
505			_a1 Optimal lower bound on eigenvector overlaps for non-Hermitian random matrices
505			_a2 Gaussian fluctuations in the equipartition principle for Wigner matrices
505			_a3 Eigenstate thermalisation at the edge for Wigner matrices
505			_a4 Out-of-time-ordered correlators for Wigner matrices
505			_a5 Eigenvector decorrelation for random matrices
505			_a6 Cusp universality for correlated random matrices
505			_a7 Prethermalization for deformed Wigner matrices
505			_a8 Loschmidt echo for deformed Wigner matrices
505			_a9 Eigenstate thermalization hypothesis for translation invariant spin systems
505			_aII BCS Theory
505			_a10 The BCS critical temperature at high density
505			_a11 The BCS energy gap at high density
505			_a12 Universality in low-dimensional BCS theory
505			_a13 Universal behavior of the BCS energy gap
505			_a14 Multi-band superconductors have enhanced critical temperatures
505			_aIII Quantum Lattice Systems
505			_a15 Local stability of ground states in locally gapped and weakly interacting quantum spin systems
505			_a16 On adiabatic theory for gapped fermionic lattice systems
505			_a17 Response theory for locally gapped systems
505			_aAppendix: Miscellaneous Results
505			_aA Deformational rigidity of Liouville metrics on the torus
505			_aB Creation rate of Dirac particles at a point source
505			_aC How a Space-Time Singularity Helps Remove the Ultraviolet Divergence Problem
505			_aBibliography
520			_aThis thesis deals with several different models for complex quantum mechanical systems and is structured in three main parts. In 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. In 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. In 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. In 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. Additionally, 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).
856			_uhttps://doi.org/10.15479/AT-ISTA-19540
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999			_c768064 _d768064