Beyond thermalization: Exploring the microscopic origin of non-thermal behavior in many-body quantum states

• Applicant: Dr. Tobias Helbig
• Subject Area: Theoretical Condensed Matter Physics ; Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 537357978

The eigenstate thermalization hypothesis (ETH) describes a mechanism how thermalization properties emerge on the quantum level. Despite extensive numerical testing, a rigorous proof of the ETH had remained elusive. In a joint work with T. Hofmann, R. Thomale and M. Greiter during my PhD, we proved the validity of the ETH in a spin-1/2 lattice with random local couplings and random on-site magnetic fields. Our combined analytical and numerical analysis provides a derivation of quantum thermalization in this setup without resorting to the standard concepts of ergodicity or maximal entropy in statistical physics nor to typical characteristics of quantum states. For that, we employed Dyson-Brownian motion random matrix theory for random banded matrices as well as the fact that the eigenvalue density of the quantum system is given by a Gaussian distribution under very general circumstances. The ETH breaks down in certain quantum systems that do not exhibit thermalization. This can for example occur in setups that are either fully integrable or display emergent integrability through many-body localization (MBL), where eigenstates are not extended but localized in regions of real space. While MBL is primarily studied in interacting systems with an added random potential, I propose to investigate a setting which also contains random couplings allowing me to extract generic properties of eigenstates from a numerical study as well as an analytical approach through random matrix theory. In view of the available theory of quantum thermalization, I investigate the microscopic origin of failure of thermalization in systems with localized many-body eigenstates in one and two spatial dimensions. Once I formulated an understanding of the non-thermal phase in spin lattices, I address the MBL-to-thermal phase transition to resolve how this transition occurs at the level of the wave function. This is accompanied by an analysis of the presence and location of a many-body mobility edge, possibly fundamental to understanding the phase transition, as well as the exploration of the role of rare regions, which might form when taking the transition from localized to thermal behavior. To diagnose thermal or non-thermal behavior, I use the numerical techniques of exact diagonalization and of the density matrix renormalization group complemented by measures of participation, the Kullback-Leibler divergence, spectral level statistics and entanglement properties such as a study of entanglement entropy obtained from post-processing the exact eigenstates and eigenvalues. The analytical analysis of the non-thermal phase as well as the phase transition is based on random matrix theory extended to incorporate local couplings. It allows for a deeper understanding of generic many-body quantum states in the localized regime, that have potential applications as reliable storage of quantum information.

• DFG Programme: WBP Fellowship
• International Connection: USA
• Host: Professor Dr. Srinivas Raghu