Critical quantum dynamics of observables at eigenstate transitions

S. Jiricek,1 M. Hopjan,2 P. Łydżba,3 F. Heidrich-Meisner,1 L. Vidmar2,4

1Institut für Theoretische Physik, Georg-August-Universität Goettingen, D-37077 Goettingen, Germany

2Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia

3Department of Theoretical Physics, Wroclaw University of Science and Technology, 50-370 Wroclaw, Poland

4Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia

It is an outstanding goal to unveil the fingerprints of universal quantum dynamics at eigenstate transitions. Focusing on quadratic fermionic Hamiltonians, we identify physical observables that exhibit critical behavior at the transition. Our result is based on two ingredients: (a) A relationship between the observable time evolution in a many-body state and the transition probabilities in single-particle states, and (b) a scale invariance of transition probabilities, which generalizes the recent result for survival probabilities [1,2]. We then show that these properties give rise to a critical behavior in the quantum quench dynamics of observables, which share the common eigenbasis with the Hamiltonian before the quench. We numerically demonstrate this phenomenon at the localization transition in the three-dimensional Anderson model, for which the critical bahavior can be detected in experimentally relevant observables such as site occupations and particle imbalance [3].

1. M. Hopjan and L.Vidmar, Phys. Rev. Lett. 131, 060404 (2023)
2. M. Hopjan and L.Vidmar, arXiv:2309.16005 (2023)
3. S. Jiricek et al., in preparation