Microscopic Foundations of the Eigenstate Thermalization Hypothesis
Theoretical Condensed Matter Physics
Final Report Abstract
This project aimed at gaining a better understanding of the eigenstate thermalization hypothesis (ETH), which has become a new cornerstone in the foundations of quantum statistical mechanics during the past decade. One of the remarkable consequences of ETH is that for generic non-integrable many-body Hamiltonians, away from the ground state single eigenstates are typical in the sense that nearby eigenstates do not convey new information (in the sense of this information being accessible by measuring few-body observables). This is a deep and important result for spectra of generic many-body problems that deserves careful scrutiny. However, to date the validity of ETH rests solely on analytical arguments for special classes of Hamiltonians or numerical data. In this project we made a step towards an analytical understanding of ETH by showing that random matrix arguments can be applied to non-integrable many-particle Hamiltonians without intrinsic randomness. To be precise, we found this to be true in all the models that we analyzed. Essentially we showed in the exemplary cases studied by us that one can transform the original Hamiltonian to a basis where its off-diagonal matrix elements appear to be random, while few-body observables still retain their few-body character. In this basis one can then apply analytical arguments due to Deutsch, and following his arguments establish that ETH is fulfilled. This work triggers numerous possibilities for follow-up projects connected to other topical problems like many-body localization or the SYK model. These follow-up projects can build on the numerical routines developed and tested above.
