We show that the transition from regular to chaotic spectral statistics in interacting many-body quantum systems has an unambiguous signature in the distribution of Schmidt coefficients dynamically generated from a generic initial state, and thus limits the efficiency of the time-dependent density-matrix renormalization-group algorithm. We investigate this mechanism on the tilted Bose-Hubbard model; however the emergence of universal spectral properties allows translation of our conclusions to generic many-body quantum systems.
One-dimensional Bose-Hubbard models are well known to obey a transition from regular to quantum-chaotic spectral statistics. We are extending this concept to relatively simple two-dimensional many-body models. Also in two dimensions a transition from regular to chaotic spectral statistics is found and discussed. In particular, we analyze the dependence of the spectral properties on the bond number of the two-dimensional lattices and the applied boundary conditions. For maximal connectivity, the systems behave most regularly in agreement with the applicability of mean-field approaches in the limit of many nearest-neighbor couplings at each site.
We numerically study the spectral statistics of open quantum many-body systems (OQMBS) as signatures of quantum chaos (or the lack thereof), using the dissipative spectral form factor (DSFF), a generalization of the spectral form factor to complex spectra. We show that the DSFF of chaotic OQMBS generically displays the
quadratic ramp-plateau behaviour of the Ginibre ensemble from random matrix theory, in contrast to the linear ramp-plateau behaviour of the Gaussian ensemble in closed quantum systems. Furthermore, in the presence of many-body interactions, such RMT behaviour emerges only after a time scale
τdev, which generally increases with system size for sufficiently large system size, and can be identified as the non-Hermitian analogue of the
many-body Thouless time. The universality of the random matrix theory behavior is demonstrated by surveying twelve models of OQMBS, including random Kraus circuits (quantum channels) and random Lindbladians (Liouvillians) in several symmetry classes, as well as Lindbladians of paradigmatic models such as the Sachdev-Ye-Kitaev (SYK), XXZ, and the transverse field Ising models. We devise an unfolding and filtering procedure to remove variations of the averaged density of states which would otherwise hide the universal RMT-like signatures in the DSFF for chaotic OQMBS. Beyond chaotic OQMBS, we study the spectral statistics of non-chaotic OQMBS, specifically the integrable XX model and a system in the many-body localized (MBL) regime in the presence of dissipation, which exhibit DSFF behaviours distinct from the ramp-plateau behaviour of random matrix theory. Lastly, we study the DSFF of Lindbladians with the Hamiltonian term set to zero, i.e. only the jump operators are present, and demonstrate that the results of RMT universality and scaling of many-body Thouless time survive even without coherent evolution.
In this thesis, we study interacting ultracold bosons in tilted optical lattices, a paradigm for complex many-body quantum systems. Our analysis is based on the tilted Bose-Hubbard Hamiltonian which incorporates inter-site tunneling, inter-particle interaction, and a static field. The competition between the three contributions induces the system’s complexity which, on the spectral level, is reflected in chaotic level statistics. We address the question how this spectral chaos influences the dynamical behavior of the system and, vice versa, which information we can infer from the system’s dynamics on its many-body spectrum. Employing methods from random matrix theory, we first perform a detailed analysis of the Hamiltonian’s regular and chaotic spectral regimes and determine their borders by means of a scaling analysis. In the main part of the thesis, we tackle the question raised above from three different perspectives: Firstly, we investigate how the Hamiltonian’s level statistics affects the simulability of the system and find a breakdown of the simulation efficiency at the crossover from the regular to the chaotic regime. Secondly, we analyze the stability of the Hamiltonian’s eigenstates within the chaotic spectral regime and identify robust states that are dynamically stable against perturbations. Thirdly, we study the fast and irreversible decay of the Bloch oscillations in the chaotic regime. We develop an approach that is based on a statistical analysis of the velocity operator and find a characteristic crossover in the Bloch oscillations’ decay, once again right at the transition from regular to chaotic level statistics.
There is a newly emerging understanding that in the chaotic domain of isolated finite interacting many particle systems smoothed densities define the statistical description of these systems and these densities follow from embedded (two-body) random matrix ensembles and their various deformations. These ensembles predict that the smoothed form of matrix elements of a transition operator between the chaotic eigenstates weighted by the densities at the two ends (i.e. the bivariate strength density) will be a bivariate Gaussian with the bivariate correlation coefficient arising out of the non-commutability of the hamiltonian and the transition operator involved. The ensemble theory extends to systems with a mean-field and a chaos generating two-body interaction (as in nuclei, atoms and diffusive quantum dots). These developments in many-body quantum chaos are described with special reference to one-body transition operators.