Random matrix theory (RMT) is a mathematical framework that has found profound applications in physics, particularly in the study of many-body systems. Its success lies in its ability to predict universal statistical properties of complex systems, independent of the specific details. This thesis explores the application of RMT to two classes of many-body systems: quantum and stochastic many-body systems. Within the quantum framework, this work focuses on the Bose-Hubbard system, which is paradigmatic for modeling ultracold atoms in optical traps. According to RMT and the Eigenstate Thermalization Hypothesis (ETH), eigenstate-to-eigenstate fluctuations of expectation values of local observables decay rapidly with the system size in the thermodynamic limit at sufficiently large temperatures. Here, we study these fluctuations in the classical limit of fixed lattice size and increasing boson number. We find that the fluctuations follow the RMT prediction for large system sizes but deviate substantially for small lattices. Partly motivated by these results, the Bose-Hubbard model on three sites is studied in more detail. On few sites, the Bose-Hubbard model is known to be a mixed system, being neither fully chaotic nor integrable. We compare energy-resolved classical and quantum measures of chaos, which show a strong agreement. Deviations from RMT predictions are attributed to the mixed nature of the few-site model. In the context of stochastic systems, generators of Markov processes are studied. The focus is on the spectrum. We present results from two investigations of Markov spectra. First, we investigate the effect of sparsity on the spectrum of random generators. Dense random matrices previously used as a model for generic generators led to very large spectral gaps and therefore to unphysically short relaxation times. In this work, a model of random generators with adjustable sparsity — number of zero matrix elements — is presented, extending the dense framework. It is shown that sparsity leads to longer, more physically realistic relaxation times. Second, the generator spectrum of the Asymmetric Simple Exclusion Process (ASEP), a quintessential model in non-equilibrium statistical mechanics, is analyzed. We investigate the spectral boundary, which is characterized by pronounced spikes. The emergence of these spikes is analyzed from several points of view, including RMT. The results presented in this thesis contribute to the understanding of the applicability of RMT to many-body systems. This thesis highlights successes such as the explanation of “ETH fluctuations” in Bose-Hubbard models, the improvement of random matrix descriptions by introducing sparsity, and the emergence of spikes in the spectral boundary of the ASEP. The latter is a notable case where RMT provides insights even though the ASEP is a Bethe-integrable system. Furthermore, this thesis shows examples of the limits of RMT, exemplified by the results presented for the Bose-Hubbard model with a few sites.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第25111号 / 理博第5018号 / 新制||理||1715(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)准教授 福間 將文, 教授 杉本 茂樹, 教授 橋本 幸士 / 学位規則第4条第1項該当 / Doctor of Agricultural Science / Kyoto University / DFAM

The first part of the present Phd thesis is devoted to transport investigations in disordered quantum systems. We aim at quantitatively determining transport parameters like conductivity, mean free path, etc., for simple models of spatially disordered and/or percolated quantum systems in the limit of high temperatures and low fillings using linear response theory. We find the transport behavior for some models to be in accord with a Boltzmann equation, i.e., long mean free paths, exponentially decaying currents although there are no band-structures to start from, while this does not apply to other models even though they are also almost completely delocalized. The second part of the present PhD thesis addresses the issue of initial state independence (ISI) in closed quantum system. The relevance of the eigenstate thermalization hypothesis (ETH) for the emergence of ISI equilibration is to some extent addressed. To this end, we investigate the Heisenberg spin-ladder and check the validity of the ETH for the energy difference operator by examining the scaling behavior of the corresponding ETH-fluctuations, which we compute using an innovative numerical method based on typicality related arguments. While, the ETH turns out to hold for the generic non-integrable models and may therefore serve as the key mechanism for ISI for this cases, it does not hold for the integrable Heisenberg-chain. However, close analysis on the dynamic of substantially out-of-equilibrium initial states indicates the occurrence of ISI equillibration in the thermodynamic limit regardless of whether the ETH is violated. Thus, we introduce a new parameter $v$, which we propose as an alternative of the ETH to indicate ISI equillibration in cases, in which the ETH does not strictly apply.