A note on weak eigenstate thermalization and normalization of operators

P. Łydżba,1 R. Świętek,2,3 M. Mierzejewski,1 M. Rigol,4 L. Vidmar2,3

1Institute of Theoretical Physics, Faculty of Fundamental Problems of Technology, Wrocław

University of Science and Technology, 50-370 Wrocław, Poland

2Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-

1000 Ljubljana, Slovenia

3Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia

4Department of Physics, The Pennsylvania State University, University Park, Pennsylvania

16802, USA

While the eigenstate thermalization hypothesis (ETH) is well established for quantum-chaotic interacting systems, its validity for other classes of systems remains a matter of intense debate. Focusing on quadratic fermionic Hamiltonians, we here argue that the weak ETH is satisfied for few-body observables in many-body eigenstates of quantum-chaotic quadratic (QCQ) Hamiltonians. In contrast, the weak ETH is violated in two cases: (a) for sums of few-body observables in all quadratic Hamiltonians, and (b) for few-body observables in localized quadratic Hamiltonians. We argue that these properties can be traced back to the validity of single-particle eigenstate thermalization, and we highlight the subtle role of normalization of operators. Our results suggest that the difference between weak and no ETH in many-body eigenstates allows for a distinction between single-particle quantum chaos and localization. We test to which degree this phenomenology holds true for integrable systems such as the XYZ and XXZ models.