Modeling complex quantum systems: Random matrices, BCS theory, and quantum lattice systems Volume 1 of 2

Thesis

Abstract

Acknowledgements

About the Author

List of Collaborators and Publications

Table of Contents

List of Figures

List of Tables

Introduction and Summary of Results

I Random Matrices

1 Optimal lower bound on eigenvector overlaps for non-Hermitian random matrices

2 Gaussian fluctuations in the equipartition principle for Wigner matrices

3 Eigenstate thermalisation at the edge for Wigner matrices

4 Out-of-time-ordered correlators for Wigner matrices

5 Eigenvector decorrelation for random matrices

6 Cusp universality for correlated random matrices

7 Prethermalization for deformed Wigner matrices

8 Loschmidt echo for deformed Wigner matrices

9 Eigenstate thermalization hypothesis for translation invariant spin systems

II BCS Theory

10 The BCS critical temperature at high density

11 The BCS energy gap at high density

12 Universality in low-dimensional BCS theory

13 Universal behavior of the BCS energy gap

14 Multi-band superconductors have enhanced critical temperatures

III Quantum Lattice Systems

15 Local stability of ground states in locally gapped and weakly interacting quantum spin systems

16 On adiabatic theory for gapped fermionic lattice systems

17 Response theory for locally gapped systems

Appendix: Miscellaneous Results

A Deformational rigidity of Liouville metrics on the torus

B Creation rate of Dirac particles at a point source

C How a Space-Time Singularity Helps Remove the Ultraviolet Divergence Problem

Bibliography

This 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).