Correlations Between Matrix Elements of Few-Body Observables in the Energy Eigenbases of Many-Body Systems

• Applicant: Professor Dr. Jochen Gemmer
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics ; Theoretical Condensed Matter Physics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 531128043

In the 1950's E. Wigner suggested to use random matrices to model the Hamiltonians of heavy nuclei. In the 1990's M. Srednicki conjectured that matrices corresponding to few body-observables displayed in the energy eigenbasis of chaotic many-body systems, had many similarities with random matrices. This claim is the essence of the (off-diagonal) "eigenstate thermalization hypothesis" (ETH). While a clean mathematical definition and a proof of the ETH are still under debate, various aspects of it have been extensively tested numerically. For example the relative frequency of values of elements of pertinent matrices has often been found to be in excellent agreement with that of random matrices. These "relative frequency investigations" are, however, unable to unveil correlations among matrix elements. More recently, other investigations that actually target these correlations, have moved to the focus of the field. The physical significance of these correlations is more subtle than that of the relative frequency of values of matrix elements: While simple 2-point temporal correlation functions do not depend on these correlations, more involved higher order correlation functions as appearing e.g. in "Out of Time Ordered Correlators" etc. certainly do. This applies as well to evolutions of expectation values entailed by initial states far from equilibrium. Hence, detecting and describing these correlations between matrix elements as well as scrutinizing their impact on relevant dynamics is a timely endeavor that we intend to tackle. It is widely accepted that the absence of correlations between matrix elements can, if at all, only apply to elements very close to the diagonal of the matrix. This regime corresponds to small frequencies which in turn correspond to long time phenomena. Thus, when approaching the question of "correlation-free submatrices" from the numerical side, it is obvious that this approach hinges on the ability to numerically observe very long time scales in many-body quantum systems. So-called Chebyshev integrators embody a competitive way to do so. The possibility to replace the unitary time propagation of matrices by the simpler time propagation of quantum states is also numerically very beneficial. This replacement is implemented to very good approximation by a concept known as quantum typicality. The group of J. Gemmer has built a proven expertise to use these numerical tools (Chebyshev integrators and quantum typicality) over the last decade, lately also and especially through the contribution of Dr. Jiaozi Wang. For the assessment of this proposal it is thus important to know that Dr. Wang will work on this project in the group of J. Gemmer. The success of the project is furthermore fostered by established and ongoing intense collaborations with notable experts including Prof. Anatoly Dymarsky, Prof. Tomaz Prosen, Prof. Giuliano Benenti, Dr. Jonas Richter and Prof. Robin Steinigeweg.

• DFG Programme: Research Grants